Download Weakly differentiable functions on varifolds

Weakly Differentiable Functions on Varifolds
U LRICH M ENNE
A BSTRACT. The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable
varifolds with locally bounded first variation. The concept proposed here is defined by means of integration-by-parts identities
for certain compositions with smooth functions. In this class,
the idea of zero boundary values is realised using the relative
perimeter of superlevel sets. Results include a variety of Sobolev
Poincaré-type embeddings, embeddings into spaces of continuous and sometimes Hölder-continuous functions, and pointwise differentiability results both of approximate and integral
type as well as coarea formulae.
As a prerequisite for this study, decomposition properties of
such varifolds and a relative isoperimetric inequality are established. Both involve a concept of distributional boundary of a
set introduced for this purpose.
As applications, the finiteness of the geodesic distance associated with varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.
C ONTENTS
Introduction
1. Notation
2. Topological Vector Spaces
3. Distributions on Products
4. Monotonicity Identity
5. Distributional Boundary
6. Decompositions of Varifolds
7. Relative Isoperimetric Inequality
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c , Vol. 65, No. 3 (2016)
Indiana University Mathematics Journal 978
991
993
998
1002
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U LRICH M ENNE
8. Generalised Weakly Differentiable Functions
9. Zero Boundary Values
10. Embeddings into Lebesgue Spaces
11. Differentiability Properties
12. Coarea Formula
13. Oscillation Estimates
14. Geodesic Distance
15. Curvature Varifolds
Appendix. Alternate Sources
References
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I NTRODUCTION
Overview. The main purpose of this paper is to present a concept of weakly
differentiable functions on nonsmooth “surfaces” in Euclidean space with arbitrary
dimension and codimension arising in variational problems involving the area
functional. Such surfaces are modelled by rectifiable varifolds whose first variation
with respect to area is representable by integration (i.e., in the terminology of Simon [Sim83, 39.2], rectifiable varifolds with locally bounded first variation). This
includes area-minimising rectifiable currents,1 in particular, perimeter-minimising
“Caccioppoli sets,” or almost every time slice of Brakke’s mean curvature flow2
just as well as surfaces occurring in diffuse interface models3 or image restoration
models.4 The envisioned concept should be defined without reference to an approximation by smooth functions, and it should be as broad as possible to still
allow for substantial positive results.
In order to integrate well the first variation of the varifold into the concept
of weakly differentiable function, it seems necessary to provide an entirely new
notion rather than to adapt one of the many concepts of weakly differentiable
functions that have been invented for different purposes. For instance, to study
the support of the varifold as metric space with its geodesic distance, stronger
conditions on the first variation are needed (see Section 14).
Description of results.
Setup and basic results. To describe the results obtained, consider the following set of hypotheses; the notation is explained in Section 1.
General hypothesis. Suppose m and n are positive integers, m ≤ n, U is an open
subset of Rn , and V is an m-dimensional rectifiable varifold in U whose first variation
δV is representable by integration.
1See Allard [All72, 4.8 (4)].
2See Brakke [Bra78, Section 4].
3See, e.g., Hutchinson and Tonegawa [HT00] or Röger and Tonegawa [RT08].
4See, e.g., Ambrosio and Masnou [AM03].
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The study of weakly differentiable functions is closely related to the study
of connectedness properties of the underlying space or varifold. Therefore, it is
instructive to begin with the latter.
Definition (see Definition 6.2). If V is as in the general hypothesis, it is
called indecomposable if and only if there is no kV k + kδV k measurable set E such
that kV k(E) > 0, kV k(U ∼ E) > 0, and δ(V x E × G(n, m)) = (δV ) x E .
The basic theorem involving this notion is the following.
Decomposition theorem (see Remark 6.10 and Theorem 6.12). If m, n, U ,
and V are as in the general hypothesis, then there exists a countable disjointed collection G of Borel sets whose union is U such that V x E × G(n, m) is nonzero and
indecomposable and δ(V x E × G(n, m)) = (δV ) x E whenever E ∈ G.
Employing the following definition, the Borel partition G of U is required to
consist of members whose distributional V boundary vanishes and that cannot be
split nontrivially into smaller Borel sets with that property.
Definition (see Definition 5.1). If m, n, U , and V are as in the general
hypothesis and E is kV k + kδV k measurable, then the distributional V boundary
of E is defined by
V ∂E = (δV ) x E − δ(V x E × G(n, m)) ∈ D ′ (U, Rn ).
If the varifold is sufficiently regular, a version of the Gauss Green theorem can
be proven that both justifies the terminology and links the concept of boundary
to the one for currents (see Theorem 5.9 and Remark 5.10).
In the terminology of Definition 6.6 and Definition 6.9, the varifolds V x E ×
G(n, m) occurring in the decomposition theorem are components of V , and the
family {V x E × G(n, m) : E ∈ G} is a decomposition of V . However, unlike the
decomposition into connected components for topological spaces, the preceding
decomposition is nonunique in an essential way. In fact, the varifold corresponding to the union of three lines in R2 meeting at the origin at equal angles may
also be decomposed into two “Y-shaped” varifolds each consisting of three rays
meeting at equal angles (see Remark 6.13).
The seemingly most natural definition for a function to be weakly differentiable with respect to a varifold satisfying the general hypothesis would be to
require an integration-by-parts identity involving the first variation of the varifold. However, the resulting class of real-valued functions is not stable under
truncation of its members, nor does a coarea formula hold (see Remarks 8.27
and 8.32). Therefore, instead, one requires that an integration-by-parts identity
simultaneously holds for each composition of the function in question with a
smooth real-valued function whose derivative has compact support (see Definition 8.3). Whenever Y is a finite-dimensional normed vector space, the resulting
class of functions of Y -valued kV k + kδV k measurable functions is denoted by
T(V , Y ).
U LRICH M ENNE
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Whenever f belongs to that class, there exists a kV k measurable Hom(Rn , Y )valued function V D f , called the generalised weak derivative of f , which is kV k
almost uniquely determined by the integration-by-parts identity. As we define
T(V , Y ), it seems natural not to require local summability of V D f but only
Z
K∩{x : |f (x)|≤s}
|V D f | dkV k < ∞,
whenever K is a compact subset of U and 0 ≤ s < ∞; this is in analogy with
1,1
(U)” introduced by Bénilan, Boccardo, Gallouët,
the definition of the space “Tloc
Gariepy, Pierre, and Vázquez in [BBG+ 95, p. 244] for the case of Lebesgue measure (see Remark 8.19). In both cases, the letter “T” in the name of the space
stands for truncation.
Stability properties under composition (e.g., truncation) then follow readily
(see Lemmas 8.12 and 8.15). Also, it is evident that members of T(V , Y ) may
be defined on components separately (see Theorem 8.24). The space T(V , Y )
is stable under addition of a locally Lipschitzian function, but it is not closed
with respect to addition in general (see Theorem 8.20 (3), and Example 8.25). A
similar statement holds for multiplication of functions (see Theorem 8.20 (4), and
Example 8.25). Adopting an axiomatic viewpoint, consider the class of functions
that satisfies the following three conditions for a given varifold:
(1) Each function that is constant on the components of some decomposition
of the varifold belongs to the class.
(2) The class is closed under addition.
(3) The class is closed under truncation.
Then, there exists a stationary one-dimensional integral varifold in R2 such that
the associated class necessarily contains characteristic functions with nonvanishing
distributional derivative representable by integration (see Example 8.28). These
characteristic functions should not belong to a class of “weakly differentiable”
functions but rather to a class of functions of “bounded variation.” The reason
for this phenomenon is the aforementioned nonuniqueness of decompositions of
varifolds.
Much of the development of the theory rests on the following basic theorem.
Coarea formula, functional analytic form (see Remark 8.5 and Theorem 8.30).
Suppose that m, n, U , and V are as in the general hypothesis, that f ∈ T(V ), and
that E(y) = {x : f (x) > y} for y ∈ R.
Then, there holds
Z
Z
hϕ(x, f (x)), V D f (x)i dkV kx = V ∂E(y)(ϕ(·, y)) dL 1 y,
Z
ZZ
g(x, f (x))|V D f (x)| dkV kx =
g(x, y) dkV ∂E(y)kx dL 1 y
whenever ϕ ∈ D (U × R, Rn ) and g : U × R → R is a continuous function with
compact support.
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An example of such a development is provided by passing from the notion of
zero boundary values on a relatively open part G of the boundary of U for sets to
a similar notion for generalised weakly differentiable functions. For kV k + kδV k
measurable sets E such that V ∂E is representable by integration, the condition is
defined in Hypothesis 7.1. In the special case that kV k is associated with M ∩U for
some properly embedded m-dimensional submanifold M of Rn ∼((Bdry U) ∼ G)
of class 2 without boundary, it is equivalent to the situation when E is of locally
finite perimeter in M and its distributional boundary is measure-theoretically contained in U (see Example 7.2). To define such a notion for nonnegative members f of T(V , R), one requires for L 1 almost all 0 < y < ∞ that the set
E(y) = {x : f (x) > y} satisfies the corresponding zero boundary value condition.
This gives rise to the subspaces TG (V ) of T(V , R) ∩ {f : f ≥ 0} with |f | ∈
T0 (V ) whenever f ∈ T(V , Y ) (see Definition 9.1 and Remark 9.2). The space
TG (V ) satisfies useful truncation and closure properties (see Lemma 9.9 and Corollary 9.13). Moreover, under a natural summability hypothesis, the multiplication
of a member of TG (V ) by a nonnegative Lipschitzian function belongs to TG (V )
(see Theorem 9.16). Whereas the usage of level sets in the definition of TG (V )
is tailored to work nicely in the proofs of embedding results in Section 10, the
stability property under multiplication requires a more delicate proof in turn.
Embedding results and structural results. To proceed to deeper results on
functions in T(V , Y ), the usage of the isoperimetric inequality for varifolds seems
indispensable. The latter works best under the following additional hypothesis.
Density hypothesis. Suppose m, n, U , and V are as in the general hypothesis and
satisfy
Θ m (kV k, x) ≥ 1 for kV k almost all x.
The key to proving effective versions of Sobolev Poincaré type embedding
results is the following theorem.
Relative isoperimetric inequality (see Corollary 7.9). Suppose m, n, U , and V
satisfy the general hypothesis and the density hypothesis, E is kV k + kδV k measurable,
n ≤ M < ∞, 1 ≤ Q ≤ M , Λ = Γ7.8 (M), 0 < r < ∞,
kV k(E) ≤ (Q − M −1 )α(m)r m ,
kV k(E ∩ {x : Θ m (kV k, x) < Q}) ≤ Λ−1 r m ,
and A = {x : U(x, r ) ⊂ U}.
Then,
kV k(E ∩ A)1−1/m ≤ Λ(kV ∂Ek(U) + kδV k(E)),
where 00 = 0.
In the special case that V ∂E = 0, kδV k(E) = 0, and
Θ m (kV k, x) ≥ Q
for kV k almost all x,
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the varifold V x E × G(n, m) is stationary and the support of its weight measure, spt(kV k x E), cannot intersect A by the monotonicity identity and the upper
bound on kV k(E). The value of the theorem lies in quantifying this behaviour.
Much of the usefulness of the result for the purposes of the present paper stems
from the fact that values of Q larger than 1 are permitted. This allows us to effectively apply the result in neighbourhoods of points where the density function
Θ m (kV k, ·) has a value greater than or equal to 2 and is approximately continuous. The case Q = 1 is partly contained in a result of Hutchinson [Hut90, Theorem 1] that treats Lipschitzian functions.
If the set E satisfies a zero boundary value condition (see Hypothesis 7.1) on a
relatively open subset G of Bdry U , the conclusion may be sharpened by replacing
A = {x : U(x, r ) ⊂ U} by
A′ = U ∩ {x : U(x, r ) ⊂ Rn ∼ B},
where B = (Bdry U) ∼ G.
In order to state a version of the resulting Sobolev Poincaré inequalities, recall
a less-known notation from Federer’s treatise on geometric measure theory (see
[Fed69, 2.4.12]) as follows.
Definition. Suppose µ measures X and f is a µ measurable function with
values in some Banach space Y .
Then, one defines µ(p) (f ) for 1 ≤ p ≤ ∞ by the formulae
µ(p) (f ) =
Z
|f |p dµ
1/p
in case 1 ≤ p < ∞,
µ(∞) (f ) = inf{s : s ≥ 0, µ({x : |f (x)| > s}) = 0}.
In comparison to the more common notation kf kLp (µ,Y ) , the method above
puts the measure in focus and avoids iterated subscripts.
Sobolev Poincaré inequality, zero median version (see Remark 8.16, Remark
9.2, and Theorem 10.1 (1a)). Suppose 1 ≤ M < ∞. Then, there exists a positive,
finite number Γ with the following property.
If m, n, U , and V satisfy the general hypothesis and the density hypothesis, n ≤
M , Y is a finite-dimensional normed vector space, f ∈ T(V , Y ), 1 ≤ Q ≤ M ,
0 < r < ∞, E = U ∩ {x : f (x) ≠ 0},
kV k(E) ≤ (Q − M −1 )α(m)r m ,
kV k(E ∩ {x : Θ m (kV k, x) < Q}) ≤ Γ −1 r m ,

∞
if m = 1,
m
β=

if m > 1,
m−1
and A = {x : U(x, r ) ⊂ U}, then
(kV k x A)(β) (f ) ≤ Γ kV k(1) (V D f ) + kδV k(1) (f ) .
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Again, apart from extending the result to the class T(V , Y ), the main improvement upon known results such as Hutchinson [Hut90, Theorem 1] is the
applicability with values Q > 1. If f belongs to TG (V ), then A may be replaced
by A′ as in the relative isoperimetric inequality. Not surprisingly, there also exists
a version of the Sobolev inequality for members of TBdry U (V ).
Sobolev inequality (see Theorem 10.1 (2a)). Suppose 1 ≤ M < ∞. Then, there
exists a positive, finite number Γ with the following property.
If m, n, U , and V satisfy the general hypothesis and the density hypothesis, n ≤
M , f ∈ TBdry U (V ), and
then there holds
E = U ∩ {x : f (x) ≠ 0}, kV k(E) < ∞,

∞
if m = 1,
m
β=

if m > 1,
m−1
kV k(β) (f ) ≤ Γ (kV k(1) (V D f ) + kδV k(1) (f )).
For Lipschitzian functions, Sobolev inequalities were obtained—for varifolds
by Allard [All72, 7.1] and Michael and Simon [MS73, Theorem 2.1], and for
rectifiable currents that are absolutely minimising with respect to a positive, parametric integrand by Federer in [Fed75, Section 2].
Coming back to the Sobolev Poincaré inequalities, one may also establish a
version with several “medians.” The number of medians needed is controlled by
the total weight measure of the varifold in a natural way.
Sobolev Poincaré inequality, several medians (see Theorem 10.7 (1)). Suppose
1 ≤ M < ∞. Then, there exists a positive, finite number Γ with the following property.
If m, n, U , and V satisfy the general hypothesis and the density hypothesis, Y
is a finite-dimensional normed vector space, sup{dim Y , n} ≤ M , f ∈ T(V , Y ),
1 ≤ Q ≤ M , N is a positive integer, 0 < r < ∞,
kV k(U) ≤ (Q − M −1 )(N + 1)α(m)r m ,
kV k({x : Θ m (kV k, x) < Q}) ≤ Γ −1 r m ,

∞
if m = 1,
m
β=

if m > 1,
m−1
and A = {x : U(x, r ) ⊂ U}, then there is a subset Υ of Y such that 1 ≤ card Υ ≤ N
and
(kV k x A)(β) (fΥ ) ≤ Γ N 1/β (kV k(1) (V D f ) + kδV k(1) (fΥ )),
where fΥ (x) = dist(f (x), Υ) for x ∈ dmn f .
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If Y = R, the approach of Hutchinson (see [Hut90, Theorem 3]) carries
over unchanged and, in fact, yields a somewhat sharper estimate (see Theorem
10.9 (1)). If dim Y ≥ 2 and N ≥ 2, the selection procedure for Υ is more delicate.
In order to precisely state the next result, the concept of approximate tangent
vectors and approximate differentiability (see [Fed69, 3.2.16]) will be recalled.
Definition. Suppose µ measures an open subset U of a normed vector space
X , a ∈ U , and m is a positive integer.
Then, Tanm (µ, a) denotes the closed cone of (µ, m) approximate tangent vectors at a consisting of all v ∈ X such that
Θ ∗m (µ x E(a, v, ε), a) > 0
for every ε > 0,
where E(a, v, ε) = X ∩ {x : |r (x − a) − v| < ε for some r > 0}.
Moreover, if X is an inner product space, then the cone of (µ, m) approximate
normal vectors at a is defined to be
Norm (µ, a) = X ∩ {u : u • v ≤ 0 for v ∈ Tanm (µ, a)}.
If V is an m-dimensional rectifiable varifold in an open subset U of Rn , then
at kV k almost all a, T = Tanm (kV k, a) is an m-dimensional plane such that
r −m
Z
f (r −1 (x − a)) dkV kx → Θ m (kV k, a)
Z
T
f dH
m
as r → 0+,
whenever f : U → R is a continuous function with compact support. However,
at individual points the requirement that Tanm (kV k, a) forms an m-dimensional
plane differs from the condition that kV k admits an m-dimensional approximate
tangent plane in the sense of Simon [Sim83, 11.8].
Definition. Suppose µ, U , a, and m are as in the preceding definition, and
f maps a subset of X into another normed vector space Y .
Then, f is called (µ, m) approximately differentiable at a if and only if there
exist b ∈ Y and a continuous linear map L : X → Y such that
Θ m (µ x X ∼{x : |f (x) − b − L(x − a)| ≤ ε|x − a|}, a) = 0
for every ε > 0.
In this case, L| Tanm (µ, a) is unique, and it is called the (µ, m) approximate differential of f at a, denoted
(µ, m) ap D f (a).
The following notation will be convenient (see Almgren [Alm00, T.1 (9)]).
Definition. Whenever P is an m-dimensional plane in Rn , the orthogonal
projection of Rn onto P will be denoted by P♮ .
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985
The Sobolev Poincaré inequality with zero median and a suitably chosen number Q is the key to proving the following structural result for generalised weakly
differentiable functions.
Approximate differentiability (see Theorem 11.2). Suppose m, n, U , and V satisfy the general hypothesis and the density hypothesis, Y is a finite-dimensional normed
vector space, and f ∈ T(V , Y ).
Then, f is (kV k, m) approximately differentiable with
m
V D f (a) = (kV k, m) ap D f (a) ◦ Tan (kV k, a)♮
at kV k almost all a.
The preceding assertion consists of two parts. First, it yields that
V D f (a)| Norm (kV k, a) = 0
for kV k almost all a,
a property that is not required by the definition. Second, it asserts that
m
V D f (a)| Tan (kV k, a) = (kV k, m) ap D f (a)
for kV k almost all a.
This is somewhat similar to the situation for the generalised mean curvature vector
of an integral m varifold, where it was first obtained by Brakke in [Bra78, 5.8] that
its tangential component vanishes, and second by the author in [Men13, 4.8] that
its normal component is induced by approximate quantities.
Differentiability in Lebesgue spaces (see Theorem 11.4 (1)). Suppose that m,
n, U , and V satisfy the general hypothesis and the density hypothesis, that Y is a
finite-dimensional normed vector space, that f ∈ T(V , Y ) ∩ Lloc
1 (kδV k, Y ), and that
n
V D f ∈ Lloc
(kV
k,
Hom
(R
,
Y
))
.
1
If m > 1 and β = m/(m − 1), then
lim r
r →0+
−m
Z
B(a,r )
|f (x) − f (a) − V D f (a)(x − a)|
|x − a|
β
dkV kx = 0
for kV k almost all a.
The result is derived from the approximate differentiability result mainly by
means of the zero-median version of the Sobolev Poincaré inequality. Another
consequence of the approximate differentiability result is the rectifiability of the
distributional boundary of almost all superlevel sets of generalised weakly differentiable functions, which supplements the functional analytic form of the coarea
formula.
Coarea formula, measure-theoretic form (see Corollary 12.2). Suppose m, n,
U , and V satisfy the general hypothesis and the density hypothesis, f ∈ T(V , R), and
E(y) = {x : f (x) > y} for y ∈ R.
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Then, there is an L 1 measurable function W with values in the weakly topologised
space of (m − 1)-dimensional rectifiable varifolds in U such that, for L 1 almost all y ,
Tanm−1 (kW (y)k, x) = Tanm (kV k, x) ∩ ker V D f (x) ∈ G(n, m − 1),
Θ m−1 (kW (y)k, x) = Θ m (kV k, x)
for kW (y)k almost all x , and
V ∂E(y)(θ) =
Z
|V D f (x)|−1 V D f (x)(θ(x)) dkW (y)kx
for θ ∈ D (U, Rn ).
In particular, kV ∂E(y)k = kW (y)k for such y .
The proof of an appropriate form of the coarea formula was the original motivation for the author to establish the new relative isoperimetric inequality and its
corresponding Sobolev Poincaré inequalities as well as the approximate differentiability result. In this respect, notice that the proof of the measure-theoretic form of
the coarea formula could not be based on the more elementary functional analytic
form of the coarea formula in conjunction with an extension of the Gauss Green
theorem (see [Fed69, 4.5.6]) to sets whose distributional V boundary is representable by integration. In fact, for general sets E whose distributional V boundary
is representable by integration, it may happen that there is no (m− 1)-dimensional
rectifiable varifold whose weight measure equals kV ∂Ek (see Remark 12.3). This
is in contrast to the behaviour of sets of locally finite perimeter in Euclidean space.
Critical mean curvature. Several of the preceding estimates and structural
results may be sharpened in case the generalised mean curvature satisfies an appropriate summability condition.
Mean curvature hypothesis. Suppose m, n, U , and V are as in the general hypothesis and satisfy the following condition.
n
If m > 1, then for some h ∈ Lloc
m (kV k, R ),
(δV )(θ) = −
Z
h • θ dkV k
for θ ∈ D (U, Rn ).
In this case,
Z ψ will denote the Radon measure over U characterised by the condition
ψ(X) = |h|m dkV k whenever X is a Borel subset of U .
X
Clearly, if this condition is satisfied, then the function h is kV k almost equal
to the generalised mean curvature vector h(V , ·) of V . The exponent m is “critical” with respect to homothetic rescaling of the varifold. Replacing it by a slightly
larger number would entail upper semicontinuity of Θ m (kV k, ·) and the applicability of Allard’s regularity theory (see Allard [All72, Section 8]). By contrast,
replacing the exponent by a slightly smaller number would allow for examples in
which the varifold is locally highly disconnected (see the author [Men09, 1.2]).
Weakly Differentiable Functions on Varifolds
987
The importance of this hypothesis for the present considerations lies in the
fact that in the relative isoperimetric inequality, the summand “kδV k(U)” may
be dropped provided V satisfies additionally the mean curvature hypothesis with
ψ(E)1/m ≤ Λ−1 (see Corollary 7.11). As a first consequence, one obtains the
following version of the Sobolev Poincaré inequality.
Sobolev Poincaré inequality, zero median, critical mean curvature (see Remark 8.16, Remark 9.2, Theorem 10.1 (1c) and (1d)). Suppose 1 ≤ M < ∞.
Then, there exists a positive, finite number Γ with the following property.
If m, n, U , V , and ψ satisfy the general hypothesis, the density hypothesis and
the mean curvature hypothesis, n ≤ M , Y is a finite-dimensional normed vector space,
f ∈ T(V , Y ), 1 ≤ Q ≤ M , 0 < r < ∞, E = U ∩ {x : f (x) ≠ 0},
kV k(E) ≤ (Q − M −1 )α(m)r m ,
ψ(E) ≤ Γ −1 ,
kV k(E ∩ {x : Θ m (kV k, x) < Q}) ≤ Γ −1 r m ,
and A = {x : U(x, r ) ⊂ U}, then the following two statements hold:
(1) If 1 ≤ q < m, then
(kV k x A)(mq/(m−q)) (f ) ≤ Γ (m − q)−1 kV k(q) (V D f ).
(2) If 1 < m < q ≤ ∞, then
(kV k x A)(∞) (f ) ≤ Γ 1/(1/m−1/q) kV k(E)1/m−1/q kV k(q) (V D f ).
Even if Q = q = 1 and f and V correspond to smooth objects, this estimate appears to be new; at least, it seems not straightforward to derive it from
Hutchinson [Hut90, Theorem 1].
A similar result holds for m = 1 (see Theorem 10.1 (1b)). Again, if f belongs
to TG (V ), then A may be replaced by A′ . Also, appropriate versions of the Sobolev
inequality may be furnished (see Theorem 10.1 (2b)–(2d)). The same is true with
respect to the Sobolev Poincaré inequalities with several medians (see Theorem
10.7 (2)–(4) and Theorem 10.9 (2)–(4)).
Before we state the stronger differentiability properties that result from the
mean curvature hypothesis, we recall the definition of relative differential from
[Fed69, 3.1.21, 3.1.22].
Definition. Suppose X and Y are normed vector spaces, A ⊂ X , a ∈ Clos A,
and f : A → Y . Then, the tangent cone of A at a, denoted Tan(A, a), is the set
of all v ∈ X such that, for every ε > 0, there exist x ∈ A and 0 < r ∈ R with
|x − a| < ε and |r (x − a) − v| < ε. Moreover, f is called differentiable relative
to A at a if and only if there exist b ∈ Y and a continuous linear map L : X → Y
such that
|f (x) − b − L(x − a)|
→ 0 as A ∋ x → a.
|x − a|
In this case, L| Tan(A, a) is unique and denoted D f (a).
U LRICH M ENNE
988
Differentiability in Lebesgue spaces, critical mean curvature (see Theorem
11.4 (2)–(4)). Suppose m, n, U , and V satisfy the general hypothesis, the density
hypothesis, and the mean curvature hypothesis, Y is a finite-dimensional normed vector
n
space, f ∈ T(V , Y ), and V D f ∈ Lloc
q (kV k, Hom(R , Y )), with 1 ≤ q ≤ ∞.
Then, the following two statements hold:
(1) If q < m and ι = mq/(m − q), then
lim r −m
r →0+
Z
B(a,r )
|f (x) − f (a) − V D f (a)(x − a)|
|x − a|
ι
dkV kx = 0
for kV k almost all a.
(2) If m = 1 or m < q, then there exists a subset A of U with kV k(U ∼ A) = 0
such that f |A is differentiable relative to A at a with
D(f |A)(a) = V D f (a)| Tanm (kV k, a) for kV k almost all a;
in particular, Tan(A, a) = Tanm (kV k, a) for such a.
If we consider V associated with two crossing lines, it is evident that one
cannot expect a function in T(V , R) to be kV k almost equal to a continuous
function even if δV = 0 and V D f = 0. Yet, in the case where f is continuous, its
modulus of continuity may be locally estimated by its generalised weak derivative.
This estimate depends on V but not on f as formulated in the next theorem.
Oscillation estimate (see Theorem 13.1 (2)). Suppose m, n, U , and V satisfy the
general hypothesis, the density hypothesis and the mean curvature hypothesis, K is a
compact subset of U , 0 < ε < ∞, U(x, ε) ⊂ Rn ∼ U for x ∈ K , and 1 < m < q.
Then, there exists a positive, finite number Γ with the following property.
If Y is a finite-dimensional normed vector space, f : spt kV k → Y is a continuous
function, f ∈ T(V , Y ), and
κ = sup{(kV k x U(a, ε))(q) (V D f ) : a ∈ K},
then
|f (x) − f (χ)| ≤ εκ
whenever x, χ ∈ K ∩ spt kV k and |x − χ| ≤ Γ −1 .
A similar result holds for m = 1 (see Theorem 13.1 (1)). This theorem rests
on the fact that connected components of spt kV k are relatively open in spt kV k
(see Corollary 6.14 (3)). If a varifold V satisfies the general hypothesis, the density
hypothesis and the mean curvature hypothesis, then the decomposition of spt kV k
into connected components yields a locally finite decomposition into relatively
open and closed subsets whose distributional V boundary vanishes (see Corollary 6.14 (2)–(4)). Moreover, any decomposition of the varifold V will refine the
decomposition of the topological space spt kV k into connected components (see
Corollary 6.14 (1)).
Weakly Differentiable Functions on Varifolds
989
When the amount of total weight measure (“mass”) available excludes the possibility of two separate sheets, the oscillation estimate may be sharpened to yield
Hölder continuity even without assuming a priori the continuity of the function
(see Theorem 13.3).
Two applications. Despite the necessarily rather weak oscillation estimate in
the general case, this estimate is still sufficient to prove that the geodesic distance
between any two points in the same connected component of the support is finite.
Geodesic distance (see Theorem 14.2). Suppose m, n, U , and V satisfy the general hypothesis, the density hypothesis and the mean curvature hypothesis, C is a connected component of spt kV k, and a, x ∈ C . Then, there exist −∞ < b ≤ y < ∞
and a Lipschitzian function g : {υ : b ≤ υ ≤ y} → spt kV k such that g(b) = a and
g(y) = x .
The proof follows a pattern common in theory of metric spaces (see Remark
14.3).
Finally, the presently introduced notion of generalised weak differentiability
may also be used to reformulate the defining condition for curvature varifolds (see
Definition 15.4 and Remark 15.5) introduced by Hutchinson in [Hut86, 5.2.1].
Characterisation of curvature varifolds (see Theorem 15.6). Suppose m and
n are positive integers, m ≤ n, U is an open subset of Rn , V is an m-dimensional
integral varifold in U ,
m
X = U ∩ {x : Tan (kV k, x) ∈ G(n, m)},
Y = Hom(Rn , Rn ) ∩ {σ : σ = σ ∗ },
and τ : X → Y satisfies
m
τ(x) = Tan (kV k, x)♮
whenever x ∈ X.
Then, V is a curvature varifold if and only if kδV k is a Radon measure absolutely
continuous with respect to kV k and τ ∈ T(V , Y ).
The condition is readily verified to be a necessary one for curvature varifolds.
The key to proving its sufficiency is to relate the mean curvature vector of V
to the generalised weak differential of the tangent plane map τ . This may be
accomplished by means of, for instance, the approximate differentiability result
(see Theorem 11.2) applied to τ , in conjunction with the previously obtained
second order rectifiability result for such varifolds (see [Men13, 4.8]).
Possible lines of further study.
Sobolev spaces. The results obtained by the author on the area formula for the
Gauss map (see [Men12b, Theorem 3]) rest on several estimates for Lipschitzian
solutions of certain linear elliptic equations on varifolds satisfying the mean curvature hypothesis. The formulation of these estimates necessitated several ad hoc
990
U LRICH M ENNE
formulations of concepts such as zero boundary values that would not seem natural from the point of view of partial differential equations. In order to avoid
repetition, the author decided to directly build an adequate framework for these
results rather than first publish the Lipschitzian case along with a proof of the results announced in [Men12b].5 The first part of such framework is provided in the
present paper. In continuing this programme, a notion of Sobolev spaces, complete vector spaces contained in T(V , Y ) in which locally Lipschitzian functions
are suitably dense, has already been developed (see [Men16]), but is not included
here for length considerations.
Functions of locally bounded variation. It seems worthwhile to investigate to what
extent results of the present paper for weakly differentiable functions extend to a
class of “functions of locally bounded variation.” A possible definition is suggested
in Remark 8.10.
Intermediate conditions on the mean curvature. The mean curvature hypothesis
loc
may be weakened by replacing Lloc
m by Lp for some 1 < p < m. In view of
the Sobolev Poincaré type inequalities obtained for height functions by the author in [Men10, Theorem 4.4], one might seek for adequate formulations of the
Sobolev Poincaré inequalities and the resulting differentiability results for functions belonging to T(V , Y ) in these intermediate cases. This could potentially
have applications to structural results for curvature varifolds (see Remarks 15.11
and 15.12). Additionally, the case p = m − 1 seems to be related to the study of
the geodesic distance for indecomposable varifolds (see Remark 14.4).
Multiple-valued weakly differentiable functions. For convergence considerations
it appears useful to extend the concept of weakly differentiable functions to a
more general class of “multiple-valued” functions in the spirit of Moser [Mos01,
Section 4] (see Remark 8.11).
Logical prerequisites. The present paper mainly depends on results in geometric measure theory in general (see Federer [Fed69]) and on varifolds in particular (see Allard [All72], [Men09], and [Men12a, Section 4]). Concerning
functional analytic material not contained in [Fed69], reference to certain items
from Schwartz [Sch73], Bourbaki [Bou87], and Dunford and Schwartz [DS58]
is made. The results on curvature varifolds in Section 15 depend additionally on
[Men13].
Organisation of the paper. In Section 1 the notation is introduced. In
Section 2, some basic terminology and results on topological vector spaces are collected. Sections 3 and 4 contain preliminary results concerning respectively certain
distributions representable by integration and consequences of the monotonicity
identity. Section 5 introduces the concept of distributional boundary of a set with
respect to a varifold. In Section 6 the decomposition properties for varifolds are
5Had the author fully anticipated the effort needed for such a construction, he might have recon-
ciled himself with some repetition.
Weakly Differentiable Functions on Varifolds
991
established, while Section 7 contains the relative isoperimetric inequality. In Sections 8–13, the theory of generalised weakly differentiable functions is presented.
In Sections 14 and 15 the applications to the study of the geodesic distance associated to certain varifolds and to curvature varifolds are discussed briefly. Finally,
in the Appendix, alternate sources for many of the items cited from [Fed69] and
[All72] are provided.
Throughout the paper, comments are included. These are not part of the
logical line of arguments but are rather offered to the reader as a guide through
the formal presentation of the material.
1. N OTATION
Notation of Federer [Fed69] and Allard [All72] will be used throughout.
Less common symbols and terminology. The difference of sets A and B is denoted
by A ∼ B . The class of all subsets of A is denoted by 2A . The domain and image of a function f are denoted dmn f and im f . A function is univalent if and
only if it is injective. The topological boundary, closure, and interior of a set A
are denoted by Bdry A, Clos A, and Int A, respectively. If X is a metric space
metrised by ̺, A ⊂ X , and x ∈ X , then the distance of x to A is defined by
dist(x, A) = inf{̺(x, a) : a ∈ A}. Whenever f is a linear map and v belongs
to its domain, the alternate notation hv, f i for f (v) is used (see [Fed69, 1.4.1]).
Inner products, by contrast, are denoted by a • (see [Fed69, 1.7.1]). The set of
positive integers is denoted by P (see [Fed69, 2.2.6]). For the open and closed
ball with centre a and radius r , the symbols U(a, r ) and B(a,
R r ) are
R employed
(see [Fed69, 2.8.1]). For integration, the alternate notations f dµ, f (x)
R dµ x ,
and µ(f ) are employed; in this respect, µ integrability
of
f
means
that
f dµ is
R
defined in R̄ and µ summability of f means that f dµ ∈ R (see [Fed69, 2.4.2]).
Whenever X is a locally compact Hausdorff space, K (X) denotes the vector space
of continuous real-valued functions on X with compact support, and K (X)∗ denotes the vector space of all Daniell integrals on X (see [Fed69, 2.5.14, 2.5.19]).
In Definition 2.10, a topology on K (X) will be defined such that K (X)∗ becomes the dual topological vector space to K (X) (see Remark 2.11). Functions
that are k times continuously differentiable and submanifolds defined by such
functions are termed “of class k”; (see [Fed69, 3.1.11, 3.1.19]). Moreover, the
value of a distribution T at ϕ is alternately denoted by T (ϕ) and Tx (ϕ(x)) (see
[Fed69, 4.1.1]). Finally, whenever U is an open subset of Rn and m is a nonnegative integer, the space of varifolds, rectifiable varifolds, and integral varifolds
of dimension m is denoted by Vm (U), RV m (U), and IV m (U), respectively (see
Allard [All72, 3.1, 3.5]).
Modifications. If f is a relation, then f [A] = {y : (x, y) ∈ f for some x ∈ A}
whenever A is a set (see Kelley [Kel75, p. 8]). Following Almgren [Alm00, T.1 (9)],
the symbol P♮ will denote the orthogonal projection of Rn onto P whenever P
is a plane in Rn . Extending [Fed69, 2.8.16], the concept of ϕ Vitali relation
U LRICH M ENNE
992
on a metric space X will also be employed when the measure ϕ fails to be finite on bounded sets but satisfies ϕ(Ui ) < ∞ for i ∈ P for some open sets
U1 , U2 , U3 , . . . covering X . The domain of the definition of V derivative, denoted
by D(ψ, ϕ, V , x), and the usage of the notions (ϕ, V ) approximate limit and
(ϕ, V ) approximate continuity of [Fed69, 2.9.1, 2.9.12], will be extended correspondingly. Extending the definitions in [Fed69, 3.2.16], one abbreviates the
prefix (ι# µ, m) for the approximate notions of tangent vectors, normal vectors,
differentiability, and differentials to (µ, m), whenever µ measures an open subset
of a normed vector space X , ι : U → X is the inclusion map, a ∈ U , and m is
a positive integer. In a similar way, Tanm (ι# µ, a) and Norm (ι# µ, a) are abbreviated to Tanm (µ, a) and Norm (µ, a). Following Schwartz [Sch66, Chapter 3,
Section 1], the vector space D (U, Y ) is given the usual locally convex topology
(see Definition 2.13), which differs from the topology employed by Federer in
[Fed69, 4.1.1] (see Remark 2.17). Moreover, the usage of kSk for distributions S
is chosen to be in accordance with Allard [All72, 4.2], which agrees with Federer’s usage in [Fed69, 4.1.5] in most but not in all cases (see Definition 2.18 and
Remark 2.19). Extending Allard [All72, 2.5 (2)], we note that whenever M is a
submanifold of Rn of class 2 and a ∈ M , the mean curvature vector of M at a is
the unique h(M, a) ∈ Nor(M, a) such that
Tan(M, a)♮ • (D g(a) ◦ Tan(M, a)♮ ) = −g(a) • h(M, a)
whenever g : M → Rn is of class 1 and g(x) ∈ Nor(M, x) for x ∈ M . Similarly,
if we extend Allard [All72, 4.7], whenever M is an m-dimensional submanifold
of Rn of class 1, E ⊂ M , and a ∈ M , the exterior normal in M to E at a denoted
by n(M, E, a) is defined to be the unique u ∈ Sn−1 ∩ Tan(M, a) such that
m
Θ m (H
Θ (H
m
m
x E ∩ {x :(x − a) • u > 0}, a) = 0,
x(M ∼ E) ∩ {x :(x − a) • u < 0}, a) = 0
whenever such u exists, and 0 otherwise. If V is an m-dimensional varifold in
U and kδV k is a Radon measure, then the generalised mean curvature vector of V
at x is the unique h(V , x) ∈ Rn such that
(δV )(bx,r · u)
r →0+ kV k B(x, r )
h(V , x) • u = − lim
for u ∈ Rn ,
where bx,r is the characteristic function of B(x, r ); hence, x ∈ dmn h(V , ·) if
and only if the above limit exists for every u ∈ Rn (see [Men12a, p. 9]).
Additional notation. If 1 ≤ p ≤ ∞, µ is a Radon measure over a locally compact Hausdorff space X , and Y is a Banach space, then Lloc
p (µ, Y ) consists of all f
such that f ∈ Lp (µ x K, Y ) whenever K is a compact subset of X . Concerning the Besicovitch Federer covering theorem, whenever n ∈ P , the number
β (n) denotes the least positive integer with the following property (see Almgren [Alm86, p. 464]): whenever G is a family of closed balls in Rn satisfying
Weakly Differentiable Functions on Varifolds
993
sup{diam B : B ∈ G} < ∞, there exist disjointed subfamilies G1 , . . . , Gβ(n) such
that
{x : B(x, r ) ∈ G for some 0 < r < ∞} ⊂
[[
{Gi : i = 1, . . . , β (n)}.
Concerning the isoperimetric inequality, whenever m ∈ P , the smallest number
with the following property is denoted by γ (m) (see [Men09, 2.2–2.4]): if n ∈
P , m ≤ n, V ∈ RV m (Rn ), kV k(Rn ) < ∞, and kδV k(Rn ) < ∞, then
kV k(Rn ∩ {x : Θ m (kV k, x) ≥ 1}) ≤ γ (m)kV k(Rn )1/m kδV k(Rn ).
Definitions in the text. The notions of Lusin space, locally convex space and locally
convex topology, inductive system and inductive limit, and strict inductive limit of
locally convex spaces are defined in Definitions 2.1, 2.3, 2.4, and 2.9, respectively.
The topologies on K (X) and D (U, Y ) are defined in Definitions 2.10 and 2.13.
The restriction of a distribution representable by integration to a set will be defined
in Definition 2.20. In Definition 5.1, the notion of distributional boundary of a
set with respect to certain varifolds is defined. The notions of indecomposability,
component, and decomposition for certain varifolds are defined in Definition 6.2,
6.6, and 6.9, respectively. The notion of generalised weakly differentiable functions
with respect to certain varifolds and the corresponding generalised weak derivatives,
V D f , and the associated space T(V , Y ) are introduced in Definition 8.3. The
space TG (V ) is defined in Definition 9.1. Finally, the notion of curvature varifold
is explained in Definition 15.4.
A convention. Each statement asserting the existence of a positive, finite number Γ
will give rise to a function depending on the listed parameters whose name is Γx.y ,
where x.y denotes the number of the statement. This is a refinement of a concept
employed by Almgren in [Alm00].
2. TOPOLOGICAL V ECTOR S PACES
Introduction. Some basic results on topological vector spaces and Lusin spaces
are gathered here mainly from Schwartz [Sch73] and Bourbaki [Bou87].
To define and study the locally convex topologies on K (X) and D (U, Z)
(see Definitions 2.10 and 2.13) by means of the general results on Lusin spaces
of Schwartz [Sch73], it is necessary to exhibit K (X) and D (U, Z) as inductive
limits of their subspaces K (X) ∩ {f : spt f ⊂ K} and D (U, Z) ∩ {θ : spt θ ⊂ K}
corresponding to all compact subsets K of X or U , respectively. First, the relevant
definitions will be recalled.
Definition 2.1 (see [Sch73, Chapter 2, Definition 2, p. 94]). Suppose X is a
topological space. Then, X is called a Lusin space if and only if X is a Hausdorff
topological space and there exists a complete, separable metric space W and a
continuous univalent map f : W → X whose image is X .
Remark 2.2. Any subset of a Lusin space is sequentially separable.
U LRICH M ENNE
994
Definition 2.3 (see [Bou87, Chapter 2, p. 23, Definition 1]). A topological
vector space is called a locally convex space if and only if there exists a fundamental
system of neighbourhoods of 0 that are convex sets; its topology is called locally
convex topology.
Definition 2.4 (see [Bou87, Chapter 2, p. 29, Example 2]). Suppose I is a
set directed by the relation ; that is, is a reflexive, transitive relation on I such
that every subset of I with at most two elements has an upper bound. Then, an
inductive system of locally convex spaces over I is a family of locally convex spaces Ei
for i ∈ I together with continuous linear maps fj,i : Ei → Ej whenever i, j ∈ I
and i j such that
fi,i = 1Ei
for i ∈ I,
fk,i = fk,j ◦ fj,i whenever i, j, k ∈ I and i j k.
A locally convex space E together with continuous linear maps gi : Ei → E for
i ∈ I is called inductive limit in the category of locally convex spaces of the inductive
system (or the inductive limit of the locally convex spaces Ei ) if and only if
gi = gj ◦ fj,i
whenever i, j ∈ I and i j
and the following universal property is satisfied: whenever F is a locally convex
space and hi : Ei → F for i ∈ I are continuous linear maps such that
hi = hj ◦ fj,i
whenever i, j ∈ I and i j,
there exists a unique continuous linear map h : E → F satisfying
hi = h ◦ gi
whenever i ∈ I.
The definition is extended from the category of locally convex spaces to the
categories of vector spaces, topological spaces, and sets by replacing locally convex
spaces and continuous linear maps, respectively, by vector spaces and linear maps,
topological spaces and continuous maps, and sets and maps.
Remark 2.5 (see [Bou87, Chapter 2, p. 29, Example 2; Chapter 2, p. 27,
Proposition 5]). Suppose E and fi are the inductive limit in the category of vector spaces of an inductive system, Ei and fj,i , of locally convex spaces. Then,
there exists a unique locally convex topology on E such that E and fi become the
inductive limit in the category of locally convex spaces.
The locally convex topology on E is characterised by the property that, whenever F is a locally convex space and h : E → F is a linear map, h is continuous if
and only if the linear maps h ◦ gi are continuous for i ∈ I .
Remark 2.6 (see [Bou89, Chapter 1, Section 2.4, Proposition 6; Bou04,
Chapter 3, Section 7.6, Proposition 6]). Suppose E and fi are the inductive limit
in the category of sets of an inductive system, Ei and fj,i , of topological spaces.
Then, there exists a unique topology on E such that E and fi become the inductive
limit in the category of topological spaces.
Weakly Differentiable Functions on Varifolds
995
The topology on E is characterised by the property that, whenever F is a
topological space and h : E → F is a map, h is continuous if and only if the maps
h ◦ gi are continuous for i ∈ I .
Remark 2.7. If I is a set directed by , and Ei for i ∈ I together with fj,i :
Ei → Ej for i, j ∈ I with i j form an inductive system of locally convex spaces
over I , and F together with hi : Ei → F for i ∈ I form the corresponding inductive
limit in the category of locally convex spaces, and E together with gi : Ei → E for
i ∈ I form the inductive limit in the category of topological spaces, then there
exists a unique continuous map h : E → F such that hi = h ◦ gi for i ∈ I . If I
is nonempty, h maps E univalently onto F by [Bou87, Chapter 2, p. 29, Example
2]. However, h may fail to be a homeomorphism (see Remark 2.17 (3)).
Remark 2.8. The notions of category and inductive limit (also referred to as
direct limit) are employed in accordance with [ML98, p. 10, p. 67–68].
For locally convex spaces, the notion of strict inductive limit will be required.
Definition 2.9. An inductive limit of locally convex spaces is called strict if
and only if the continuous linear maps of the corresponding inductive system are
homeomorphic embeddings.
Now, the topology on the space K (X) of continuous real-valued functions on
a locally compact Hausdorff space X with compact support may be introduced.
Definition 2.10 (see [Bou87, Chapter 2, p. 29, Example 2]6). Suppose X is
a locally compact Hausdorff space. Consider the inductive system consisting of
the locally convex spaces K (X) ∩ {f : spt f ⊂ K} with the topology of uniform
convergence corresponding to all compact subsets K of X and its inclusion maps.
Then, K (X) is endowed with the unique locally convex topology so that
K (X) together with the inclusions of K (X) ∩ {f : spt f ⊂ K} into K (X) is the
inductive limit in the category of locally convex spaces (see Remark 2.5).
Remark 2.11 (see [Bou87, Chapter 2, p. 29, Example 2]). The locally convex
topology on K (X) is Hausdorff, and induces the given topology on each closed
subspace K (X) ∩ {f : spt f ⊂ K}. Moreover, the space K (X)∗ of Daniell integrals on K (X) agrees with the space of continuous linear functionals on K (X).
Remark 2.12. If K(i) is a sequence
of compact subsets of X with K(i) ⊂
S
Int K(i + 1) for i ∈ P and X = ∞
K(i)
, then K (X) is the strict inductive
i=1
limit of the sequence of locally convex spaces K (X) ∩ {f : spt f ⊂ K(i)}.
In a similar way, one may introduce a locally convex topology on D (U, Z), the
vector space of all Z -valued functions of class ∞ with compact support in U , by
considering the subspaces DK (U, Z) = D (U, Z) ∩ {θ : spt θ ⊂ K} corresponding
to all compact subsets K of U . The latter spaces are equipped with the topology
of “uniform convergence of derivatives of arbitrary order” as a subspace topology
from E (U, Z), the space of functions of class ∞ mapping U into Z .
6In the terminology of [Bou87, Chapter 2, p. 29, Example 2], a locally compact Hausdorff space
is called a locally compact space.
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U LRICH M ENNE
Definition 2.13. Suppose U is an open subset of Rn and Z is a Banach space.
Consider the inductive system consisting of the locally convex spaces DK (U, Z)
with the topology induced from E (U, Z) corresponding to all compact subsets K
of U and its inclusion maps.
Then, D (U, Z) is endowed with the unique locally convex topology so that
D (U, Z) together with the inclusions of DK (U, Z) into D (U, Z) is the inductive
limit in the category of locally convex spaces (see Remark 2.5).
The first two remarks are analogous to those concerning K (X).
Remark 2.14. The locally convex topology on D (U, Z) is Hausdorff, and
induces the given topology on each closed subspace DK (U, Z).
Remark 2.15. If K(i) is a sequence
of compact subsets of U with K(i) ⊂
S
Int K(i + 1) for i ∈ P and U = ∞
K(i)
, then D (U, Z) is the strict induci=1
tive limit of the sequence of locally convex spaces DK(i) (U, Z). In particular, the
convergent sequences in D (U, Z) are precisely the convergent sequences in some
DK (U, Z) (see [Bou87, Chapter 3, p. 3, Proposition 2; Chapter 2, p. 32, Proposition 9 (i) (ii); Chapter 3, p. 5, Proposition 6]).
Next, we shall summarise how the definition of the topology on D (U, Z)
would be affected by taking the inductive limit in the category of topological
spaces.
Remark 2.16. The locally convex topology on D (U, Z) is the finest topology
compatible with the vector space structure of D (U, Z) such that the inclusions of
DK (U, Z) into D (U, Z) are continuous by Remark 2.15 and [Bou87, Chapter 2,
p. 75, Exercise 14].
Remark 2.17. Consider the inductive system consisting of the topological
spaces DK (U, Z) corresponding to all compact subsets K of U and its inclusion
maps, and let S denote the unique topology so that D (U, Z) together with the
inclusions of DK (U, Z) into D (U, Z) is the inductive limit in the category of
topological spaces (see Remark 2.6). Denote the topology on D (U, Z) described
in Definition 2.13 by T .
Then, the following three statements hold:
(1) Amongst all locally convex topologies on D (U, Z), the topology T is characterised by the following property: a linear map from D (U, Z) into some
locally convex space is T continuous if and only if it is S continuous.
(2) The S closed sets are precisely the T sequentially closed sets (see Remark
2.15).
(3) If U is nonempty and dim Z > 0, then S is strictly finer than T ; compare
Shirai [Shi59, Théorème 5].7 In particular, in this case S is not compatible
with the vector space structure of D (U, Z) by Remark 2.16.
7The topological preliminaries of [Shi59] are quite general. For the purpose of verifying only the
cited result, one may replace the terms “topologie” (respectively, “véritable-topologie”) with operators
satisfying conditions (a), (b), and (d) (respectively, all) of the Kuratowski closure axioms (see [Kel75,
p. 43]). (Extending [Shi59, Théorème 5] that treats the case U = Rn and Z = R to the case stated
poses no difficulty.)
Weakly Differentiable Functions on Varifolds
997
Note that, before proceeding to deeper properties of the topologies of K (X)
and D (U, Z) and their duals, the necessary terminology for distributions is completed. First, a variation measure is defined.
Definition 2.18. Suppose U is an open subset of Rn , Z is a separable Banach
space, and S : D (U, Z) → R is linear. Then, kSk is defined to be the largest Borel
regular measure over U such that
kSk(G) = sup{S(θ) : θ ∈ D (U, Z) with spt θ ⊂ G and |θ(x)| ≤ 1 for x ∈ U}
whenever G is an open subset of U .8
Only finite-dimensional spaces Z will be employed in D (U, Z) in this paper.
Remark 2.19. This concept agrees with [Fed69, 4.1.5] in the case where kSk
is a Radon measure (which is equivalent to when S is a distribution in U of type Z
representable by integration), in which case S(θ) continues to denote the value of
the unique kSk(1) continuous extension of S to L1 (kSk, Z) at θ ∈ L1 (kSk, Z). In
general, the concepts differ, since it may happen that |S(θ)| > kSk(|θ|) for some
θ ∈ D (U, Z); an example is provided by taking U = Z = R and S = ∂(δ0 ∧ e1 )
(see [Fed69, 4.1.15]).
Next, for distributions representable by integration, the concept of restriction
to a measurable set is introduced.
Definition 2.20. Suppose U is an open subset of Rn , Z is a separable Banach
space, S ∈ D ′ (U, Z) is representable by integration, and A is kSk measurable.
Then, the restriction of S to A, S x A ∈ D ′ (U, Z), is defined by
(S x A)(θ) = S(θA )
whenever θ ∈ D (U, Z),
where θA (x) = θ(x) for x ∈ A and θA (x) = 0 for x ∈ U ∼ A.
Remark 2.21. This extends the notion for currents in [Fed69, 4.1.7].
The following basic theorem on strict inductive limits of an increasing sequence of Fréchet spaces allows us to readily deduce all properties of the topologies
on K (X) and D (U, Z), and their duals necessary for the development in the subsequent sections. Notice that a Borel family is termed σ -algebra in [Sim83, 1.1].
Theorem 2.22. Suppose E is the strict inductive limit of an increasing sequence of
locally convex spaces Ei ; suppose also the spaces Ei are separable, complete (with respect
to their topological vector space structure), and metrisable for i ∈ P ,9 and the dual E ′
of E is endowed with the compact open topology.
Then, E and E ′ are Lusin spaces, and whenever D is a dense subset of E the Borel
family of E ′ is generated by the family
{E ′ ∩ {u : u(x) < t} : x ∈ D and t ∈ R}.
8Existence may be shown by use of the techniques occurring in [Fed69, 2.5.14].
9That is, E are separable Fréchet spaces in the terminology of [Bou87, Chapter 2, p. 24].
i
998
U LRICH M ENNE
❐
Proof. First, note that Ei is a Lusin space for i ∈ P , since it may be metrised
by a translation-invariant metric by [Bou87, Chapter 2, p. 34, Proposition 11].
Second, note that E is Hausdorff and induces the given topology on Ei by [Bou87,
Chapter 2, p. 32, Proposition 9 (i)]. Therefore, E is a Lusin space by [Sch73,
Chapter 2, Corollary 2 to Theorem 5, p. 102]. Since every compact subset K of
E is a compact subset of Ei for some i by [Bou87, Chapter 3, p. 3, Proposition
2; Chapter 2, p. 32, Proposition 9 (ii); Chapter 3, p. 5, Proposition 6], it follows
that E ′ is a Lusin space from [Sch73, Chapter 2, Theorem 7, p. 112]. Defining
the continuous, univalent map ι : E ′ → RD by ι(u) = u|D for u ∈ E ′ , the initial
topology on E ′ induced by ι is a Hausdorff topology coarser than the compact
open topology; hence, their Borel families agree by [Sch73, Chapter 2, Corollary 2
to Theorem 4, p. 101].
In the present context, only the weaker assertion resulting from replacement
of the compact open topology with the weak topology in the preceding theorem
will be employed. Recall here that K (X)+ = K (X) ∩ {f : f ≥ 0}.
Example 2.23. Suppose X is a locally compact Hausdorff space that admits
a countable base of its topology. Then, K (X) with its locally convex topology
(see Definition 2.10) and K (X)∗ with its weak topology are Lusin spaces, and
the Borel family of K (X)∗ is generated by the sets K (X)∗ ∩ {µ : µ(f ) < t}
corresponding to f ∈ K (X) and t ∈ R. Moreover, the topological space
K (X)∗ ∩ {µ : µ is monotone}
is homeomorphic to a complete, separable metric space;10 in fact, by choosing a
countable dense subset D of K (X)+ , the image of its homeomorphic embedding
into RD is closed.
Example 2.24. Suppose U is an open subset of Rn and Z is a separable Banach space. Then, D (U, Z) with its locally convex topology (see Definition 2.13)
and D ′ (U, Z) with its weak topology are Lusin spaces, and the Borel family of
D ′ (U, Z) is generated by the sets D ′ (U, Z) ∩ {S : S(θ) < t} corresponding to
θ ∈ D (U, Z) and t ∈ R. Moreover, if we recall Remark 2.2 and [Fed69, 4.1.5], it
follows that
(D ′ (U, Z) × K (U)∗ ) ∩ {(S, kSk) : S is representable by integration}
is a Borel function whose domain is a Borel set (with respect to the weak topology
on both spaces).
3. D ISTRIBUTIONS ON P RODUCTS
Introduction. The purpose of the present section is to separate functional analytic considerations from those employing properties of the varifold or the generalised weakly differentiable function in deriving the various coarea-type formulae
in Lemma 8.1, Theorem 8.30, Theorem 12.1, and Corollary 12.2.
10Such spaces are termed polish spaces in [Sch73, Chapter 2, Definition 1].
Weakly Differentiable Functions on Varifolds
999
❐
Inspection shows that the proof of the density result in [Fed69, 4.1.3] with
respect to the topology defined in [Fed69, 4.1.1] in fact yields sequential density
(which is the same for both topologies considered by Remark 2.15).
Lemma 3.1. Suppose U and V are open subsets of Euclidean spaces and Z is
a Banach space. Then, the image of D (U, R) ⊗ D (V , R) ⊗ Z in D (U × V , Z) is
sequentially dense.
Proof. One may proceed as in [Fed69, 1.1.3, 4.1.2, 4.1.3].
A basic consequence of differentiation theory for measures is as follows.
Lemma 3.2. If J is an open subset of R, R ∈ D ′ (J, R) is representable by
integration, and kRk is absolutely continuous with respect to L 1 |2J , then there exists
1 J
k ∈ Lloc
1 (L |2 ) such that
Z
R(ω) =
J
ωk dL 1
whenever ω ∈ L1 (kRk),
k(y) = lim ε−1 R(iy,ε )
ε→0+
for L 1 almost all y ∈ J,
where iy,ε is the characteristic function of the interval {υ : y < υ ≤ y + ε}.
❐
Proof. Localising the problem, one employs [Fed69, 2.5.8] to construct k
satisfying the first condition, which implies the second one by [Fed69, 2.8.17,
2.9.8].
The next lemma for use in later sections is readily deduced from basic measuretheoretic facts. Concerning the upper integral defined in [Fed69, 2.4.10], recall
Z∗
A
f dµ = inf
Z
B
f dµ : A ⊂ B and B is µ measurable
whenever A ⊂ X , µ measures X , and f is a nonnegative µ measurable function.
Lemma 3.3. Suppose n ∈ P , µ is a Radon measure over an open subset U of
Rn , J is an open subset of R, f is a µ measurable real-valued function, A = f −1 [J],
and F is a µ measurable Hom(Rn , R)-valued function with
Z
K∩{x : f (x)∈I}
|F | dµ < ∞
whenever K is a compact subset of U and I is a compact
subset of J . Furthermore,
Z
suppose that T ∈ D ′ (U × J, Rn) satisfies T (ϕ) =
ϕ ∈ D (U × J, Rn ).
Then, T is representable by integration, and
T (ϕ) =
Z
g dkT k =
Z
ZA
A
A
hϕ(x, f (x)), F (x)i dµ x for
hϕ(x, f (x)), F (x)i dµ x,
g(x, f (x))|F (x)| dµ x
whenever ϕ ∈ L1 (kT k, Rn) and g is an R̄-valued kT k integrable function.
U LRICH M ENNE
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Proof. Define p : U × J → U by
p(x, y) = x
for (x, y) ∈ U × J.
Define the measure ν over U by ν(B) =
Z∗
A∩B
|F | dµ for B ⊂ U . Let G : A → U ×J
be defined by G(x) = (x, f (x)) for x ∈ A. Employing [Fed69, 2.2.2, 2.2.3,
2.4.10] yields that ν x{x : f (x) ∈ I} is a Radon measure whenever I is a compact
subset of J , and hence so is the measure G# (ν|2A ) over U × J by [Fed69, 2.2.2,
2.2.3, 2.2.17, 2.3.5]. One deduces that
T (ϕ) =
Z
hϕ, |F ◦ p|−1 F ◦ pi dG# (ν|2A )
for ϕ ∈ D (U × J, Rn);
in fact, F |A = F ◦ p ◦ G, and hence both sides equal
Z
hϕ ◦ G, |F |−1 F i dν by
❐
[Fed69, 2.4.10, 2.4.18 (1)]. Consequently, kT k = G# (ν|2A ), and the conclusion
follows by means of [Fed69, 2.2.2, 2.2.3, 2.4.10, 2.4.18 (1)].
Note that if we employ the differentiation theory for measures, the following
disintegration-type theorem on distributions representable by integration results.
To link the differentiation of certain measures ψ on the real line occurring in its
proof to differentiation of functions, it is convenient to employ the L 1 Vitali
relation
V = {(x, I) : x ∈ I ⊂ R and I is a compact, nondegenerate interval}
and the corresponding V derivative D(ψ, L 1 , V , x) at x ∈ R, which is defined as
a suitable limit as diam I → 0 of the ratio ψ(I)/L 1 (I) corresponding to all I with
(x, I) ∈ V (see [Fed69, 2.8.16, 2.8.17, 2.9.1]).
Theorem 3.4. Suppose U is an open subset of Rn , J is an open subset of R,
Z is a separable Banach space, T ∈ D ′ (U × J, Z) is representable by integration,
Rθ ∈ D ′ (J, R) satisfy
Rθ (ω) = T(x,y) (ω(y)θ(x))
whenever ω ∈ D (J, R) and θ ∈ D (U, Z),
and S(y) : D (U, Z) → R satisfy (see Lemma 3.2)
S(y)(θ) = lim ε−1 Rθ (iy,ε ) ∈ R
ε→0+
for θ ∈ D (U, Z)
whenever y ∈ J , that is, y ∈ dmn S if and only if the limit exists and belongs to R
for θ ∈ D (U, Z). Then, S is an L 1 x J measurable function with respect to the weak
topology on D ′ (U, Z), and the following two statements hold:
Weakly Differentiable Functions on Varifolds
1001
(1) If g is an {u : 0 ≤ u ≤ ∞}-valued kT k measurable function, then
Z Z
J
1
g(x, y) dkS(y)kx dL y ≤
Z
g dkT k.
(2) If kRθ k is absolutely continuous with respect to L 1 |2J for θ ∈ D (U, Z),
then
Z
S(y)(ϕ(·, y)) dL 1 y,
Z
Z Z
g dkT k =
g(x, y) dkS(y)kx dL 1 y
T (ϕ) =
J
J
whenever ϕ ∈ L1 (kT k, Z) and g is an R̄-valued kT k integrable function.
Proof. Define p : U × J → R and q : U × J → U by
p(x, y) = x and q(x, y) = y
for (x, y) ∈ U × J.
First, one derives that S is an L 1 x J measurable function with values in
D ′ (U, Z) ∩ {Σ : Σ is representable by integration},
where the weak topology on D ′ (U, Z) is employed; in fact, choosing countable,
sequentially dense subsets C and D of K (U)+ and D (U, Z), respectively (see
Remark 2.2, Example 2.23, and Example 2.24), and noting that S(y) belongs to
the set in question whenever
lim ε−1 kT k((f ◦ p)(iy,ε ◦ q)) ∈ R,
ε→0+
lim ε−1 Rθ (iy,ε ) ∈ R
ε→0+
for f ∈ C and θ ∈ D by means of Remark 2.15, we see that the assertion follows
from [Fed69, 2.9.19] and Example 2.24.
To prove (1), one may assume g ∈ K (U × J)+ . Suppose f ∈ K (U)+ ,
h ∈ K (J)+ , and g = (fZ ◦ p)(h ◦ q) and β denotes the Radon measure over
∗
U × J defined by β(B) =
B
f ◦ p dkT k for B ⊂ U × J . Noting
kS(y)k(f ) ≤ D(q# β, L 1 |2J , V , y)
for L 1 almost all y ∈ J,
where V = {(b, I) : b ∈ I ⊂ J and I is a compact, nondegenerate interval}, and
employing [Fed69, 2.4.10, 2.8.17, 2.9.7] and Example 2.24, one infers that
Z
J
1
kS(y)k(f )h(y) dL y ≤
Z
J
D(q# β, L 1 |2J , V , y)h(y) dL 1 y
≤ (q# β)(h) = kT k(g).
U LRICH M ENNE
1002
An arbitrary g ∈ K (U × J)+ may be approximated by a sequence of functions
that are nonnegative linear combinations of functions of the previously considered
type (compare [Fed69, 4.1.2, 4.1.3]).
To prove the first equation in (2), it is sufficient to exhibit a sequentially dense
subset E of D (U × J, Z) such that
T (ϕ) =
Z
J
S(y)(ϕ(·, y)) dL 1 y
whenever ϕ ∈ E
by (1) and Remark 2.15. If θ ∈ D (U, Z), then
T(x,y) (ω(y)θ(x)) = Rθ (ω) =
Z
J
S(y)x (ω(y)θ(x)) dL 1 y
❐
for ω ∈ D (J, R) by Lemma 3.2. One may now take E to be the image of
D (J, R) ⊗ D (U, Z) in D (U × J, Z) (see Lemma 3.1).
The second equation in (2) follows from (1) and the first equation in (2).
4. M ONOTONICITY I DENTITY
Introduction. The purpose of this section is to derive the modifications and
consequences of the monotonicity identity that will be employed in Theorems
6.12, 8.34, 13.1, and 14.2.
First, an elementary calculus lemma is provided.
Lemma 4.1. Suppose that 1 ≤ m < ∞, 0 ≤ s < r < ∞, 0 ≤ κ < ∞,
I = {t : s < t ≤ r }, and f : I → {y : 0 < y < ∞} is a function satisfying
1/m
lim sup f (u) ≤ f (t) ≤ f (r ) + κf (r )
u→t−
Then,
f (t) ≤ 1 + m
−1
r
κ log
t
Zr
u−1 f (u)1−1/m dL 1 u
m
f (r )
t
for t ∈ I.
for t ∈ I.
Proof. Suppose f (r ) < y < ∞ and υ = m−1 κf (r )1/m y −1/m , and consider
the set J of all t ∈ I such that
r m
f (u) ≤ 1 + υ log
y
u
whenever t ≤ u ≤ r .
Clearly, J is an interval and r belongs to the interior of J relative to I . The same
holds for t with s < t ∈ Clos J , since
Therefore, I equals J .
❐
Zr
r m−1
dL 1 u
f (t) ≤ f (r ) + κf (r )1/m y 1−1/m
u−1 1 + υ log
u
t
r m
r m
= f (r ) +
1 + υ log
− 1 y < 1 + υ log
y.
t
t
Weakly Differentiable Functions on Varifolds
1003
Next, a version of the monotonicity identity without hypotheses on the first
variation of the varifold is derived.
Theorem 4.2. Suppose that m, n ∈ P , m ≤ n, a ∈ Rn , 0 < r < ∞,
V ∈ Vm (U(a, r )), and ̺ ∈ D ({s : 0 < s < r }, R). Then,
−
Zr
0
̺′ (s)s −m kV k B(a, s) dL 1 s
Z
=
̺(|x − a|)|x − a|−m−2 |P♮⊥ (x − a)|2 dV (x, P )
(U(a,r ) ∼{a})×G(n,m)
Z r
− (δV )x
s −m−1 ̺(s) dL 1 s (x − a) .
|x−a|
Proof. Assume a = 0 and let I = {s : −∞ < s < r } and J = {s : 0 < s < r }.
If ω ∈ E (I, R), sup spt ω < r , 0 ∉ spt ω′ and θ : U(a, r ) → Rn is associated
with ω by θ(x) = ω(|x|)x for x ∈ U(a, r ), then D θ(0) • P♮ = mω(0) and
D θ(x) • P♮ = |P♮ (x)|2 |x|−1 ω′ (|x|) + mω(|x|)
= −|P♮⊥ (x)|2 |x|−1 ω′ (|x|) + |x|ω′ (|x|) + mω(|x|)
whenever x ∈ Rn , 0 < |x| < r , and P ∈ G(n, m). Define ω, γ ∈ E (I, R) by
ω(s) = −
Zr
u−m−1 ̺(u) dL 1 u,
sup{s,0}
γ(s) = sω′ (s) + mω(s)
for s ∈ I ; hence, sup spt γ ≤ sup spt ω < r , 0 ∉ spt ω′ , and
ω′ (s) = s −m−1 ̺(s),
ω′′ (s) = −(m + 1)s −m−2 ̺(s) + s −m−1 ̺(s)
for s ∈ J . Using Fubini’s theorem, one computes with θ as before that
Z
δV (θ) +
̺(|x|)|x|−m−2 |P♮⊥ (x)|2 dV (x, P )
(Rn ×G(n,m))∩{(x,P ) : 0<|x|<r }
Z
ZZ r
= γ(|x|) dkV kx = −
γ ′ (s) dL 1 s dkV kx
|x|
Zr
=−
γ ′ (s)kV k B(a, s) dL 1 s.
Finally, notice that γ ′ (s) = sω′′ (s) + (m + 1)ω′ (s) = s −m ̺′ (s) for s ∈ J .
❐
0
Remark 4.3. This is a slight generalisation of Simon’s version of the monotonicity identity (see [Sim83, 17.3]), included here for the convenience of the
reader.
U LRICH M ENNE
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Hypothesis 4.4. Suppose that m, n ∈ P , m ≤ n, U is an open subset of
Rn , V ∈ Vm (U), kδV k is a Radon measure, and η(V , ·) is a kδV k measurable
Sn−1 -valued function satisfying
(δV )(θ) =
Z
η(V , x) • θ(x) dkδV kx
for θ ∈ D (U, Rn )
(see Allard [All72, 4.3]).
Corollary 4.5. Suppose m, n, U , V , and η are as in Hypothesis 4.4. Then,
s −m kV k B(a, s) +
Z
(B(a,r ) ∼ B(a,s))×G(n,m)
|x − a|−m−2 |P♮⊥ (x − a)|2 dV (x, P )
= r −m kV k B(a, r )
Z
(sup{|x − a|, s}−m − r −m )(x − a) • η(V , x) dkδV kx
+ m −1
B(a,r )
whenever a ∈ Rn , 0 < s ≤ r < ∞, and B(a, r ) ⊂ U .
❐
Proof. Letting ̺ approximate the characteristic function of {t : s < t ≤ r },
the assertion is a consequence of Theorem 4.2.
Remark 4.6. Using Fubini’s theorem, the last summand can be expressed as
Zr
s
t −m−1
Z
(x − a) • η(V , x) dkδV kx dL 1 t.
B(a,t)
Remark 4.7. Corollary 4.5 and Remark 4.6 are minor variations of Simon
[Sim83, 17.3, 17.4].
Properties of the density ratios, which in the case where m > 1 often require
the hypothesis p > m, may be obtained for p = m = 1 as well; here, the terminology of Hypothesis 6.1 concerning the p-th power summability of the generalised
mean curvature vector is anticipated.
Corollary 4.8. Suppose m, n, U , and V are as in Hypothesis 4.4, m = 1, and
X = U ∩ {a : kδV k({a}) > 0}.
Then, the following three statements hold:
(1) If a ∈ Rn , 0 < s ≤ r < ∞, and B(a, r ) ⊂ U , then
s
−1
kV k B(a, s) +
Z
(B(a,r ) ∼ B(a,s))×G(n,1)
|x − a|−3 |P♮⊥ (x − a)|2 dV (x, P )
≤ r −1 kV k B(a, r ) + kδV k(B(a, r ) ∼{a}).
(2) Θ 1 (kV k, ·) is a real-valued function whose domain is U .
(3) Θ 1 (kV k, ·) is upper semicontinuous at a whenever a ∈ U ∼ X .
Weakly Differentiable Functions on Varifolds
1005
If, also, Θ 1 (kV k, x) ≥ 1 for kV k almost all x , then the following two statements
hold:
(4) If a ∈ spt kV k, then
Θ 1 (kV k, a) ≥


1
1


2
if a 6∈ X,
if a ∈ X.
(5) If a ∈ spt kV k, 0 < s ≤ r < ∞, U(a, r ) ⊂ U , and kδV k(U(a, r ) ∼{a}) ≤
ε, then
kV k U(x, s) ≥ 2−1 (1 − ε)s
whenever x ∈ spt kV k and |x − a| + s ≤ r .
Proof. If a ∈ U , 0 < s < r < ∞ and B(a, r ) ⊂ U , then
| sup{|x − a|, s}−1 − r −1 | |x − a| ≤ 1
whenever x ∈ B(a, r ).
Therefore, (1) follows from Corollary 4.5. Then, (1) readily implies (2) and (3)
and the first half of (4). To prove the second half, choose η as in Hypothesis 4.4,
and consider a ∈ X . One may assume a = 0 and, in view of Allard [All72,
4.10 (2)], also U = Rn . Abbreviating v = η(V , 0) ∈ Sn−1 and defining the
reflection f : Rn → Rn by f (x) = x − 2(x • v)v for x ∈ Rn , one infers the
second half of (4) by applying the first half of (4) to the varifold V + f# V .
If a, s , r , ε, and x satisfy the conditions of (5), then
kV k U(x, s) ≥ (1 − ε)s if either s ≤ |x − a| or x = a,
s
≥ 2−1 (1 − ε)s if 2|x − a| ≤ s
kV k U(x, s) ≥ kV k U a,
by (1) and (4), and the case |x − a| < s < 2|x − a| then follows.
❐
2
Finally, a corollary for use in case p = m ≥ 2 is formulated; here, the terminology of Hypothesis 6.1 is employed again.
Corollary 4.9. Suppose m, n, U , and V are as in Hypothesis 4.4, a ∈ Rn ,
0 < s < r < ∞, B(a, r ) ⊂ U , 0 ≤ κ < ∞, and
kδV k B(a, t) ≤ κr −1 kV k(B(a, r ))1/m kV k(B(a, t))1−1/m
for s < t < r ,
where 00 = 1. Then,
r m −m
r
kV k B(a, r ).
s −m kV k B(a, s) ≤ 1 + m−1 κ log
s
the conclusion.
❐
Proof. Assume kV k B(a, r ) > 0 and define t = inf{u : kV k B(a, u) > 0} and
f (u) = u−m kV k B(a, u) for sup{s, t} < u ≤ r . Then, in view of Corollary 4.5
and Remark 4.6, one may apply Lemma 4.1 with s replaced by sup{s, t} to infer
U LRICH M ENNE
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5. D ISTRIBUTIONAL B OUNDARY
Introduction. In this section, the notion of distributional boundary of a set
with respect to certain varifolds is introduced (see Definition 5.1). Moreover, a
basic structural theorem is proven (see Theorem 5.9) that allows us to compare
this notion to a similar one employed by Bombieri and Giusti in the context of
area minimising currents in [BG72, Theorem 2] (see Remark 7.14).
First, the definition of distributional boundary is given.
Definition 5.1. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn ,
V ∈ Vm (U), kδV k is a Radon measure, and E is kV k + kδV k measurable. Then,
the distributional V boundary of E is given by (see Definition 2.20)
V ∂E = (δV ) x E − δ(V x E × G(n, m)) ∈ D ′ (U, Rn ).
Remark 5.2. If W ∈ Vm (U), kδW k is a Radon measure and E is additionally
kW k + kδW k measurable, then
(V + W ) ∂E = V ∂E + W ∂E.
Remark 5.3. If E and F are kV k + kδV k measurable sets and E ⊂ F , then
V ∂(F ∼ E) = (V ∂F ) − (V ∂E).
Remark 5.4. If V ∂E is representable by integration, W = V x E × G(n, m),
and F is a Borel set, then
V ∂(E ∩ F ) = W ∂F + (V ∂E) x F .
Example 5.5. Suppose E and P are distinct members of G(2, 1), and V ∈
IV 1 (R2 ) is characterised by kV k = H 1 x(E ∪ P ).
Then, δV = 0 and V ∂E = 0, but there exists no sequence of locally Lipschitzian functions fi : R2 → R satisfying
Z
|fi − f | + |(kV k, 1) ap D fi | dkV k → 0
as i → ∞,
where f is the characteristic function of E .
Remark 5.6. Since it will follow from Theorem 8.24 (3) that f is a generalised weakly differentiable function with vanishing generalised weak derivative,
the preceding example shows that the theory of generalised weakly differentiable
functions cannot be developed by using approximation by locally Lipschitzian
functions. Instead, the relevant properties of sets are studied first in Sections 5–7
before proceeding to the theory of generalised weakly differentiable functions in
Sections 8–13.
Weakly Differentiable Functions on Varifolds
1007
In order to treat exceptional sets, a lemma concerning capacity is recorded.
Lemma 5.7. Suppose n ∈ P , 1 ≤ m ≤ n, U is an open subset of Rn , µ is a
Radon measure over U , K is a compact subset of U , A is a compact subset of Int K ,
H m−1 (A) = 0, and
lim sup{s −m µ B(x, s) : x ∈ A and 0 < s ≤ r } < ∞.
r →0+
Then, there exists a sequence fi ∈ E (U, R) satisfying
0 ≤ fi ≤ 1, A ⊂ Int{x : fi (x) = 0}, {x : fi (x) < 1} ⊂ K for i ∈ P ,
lim fi (x) = 1 for H m−1 almost all x ∈ U,
i→∞
lim
i→∞
Z
| D fi | dµ = 0.
Proof. Let ϕ∞ denote the size ∞ approximating measure for H m−1 over Rn .
Observe that it is sufficient to prove that for ε > 0 there exists f ∈ E (U, R) with
0 ≤ f ≤ 1,
A ⊂ Int{x : f (x) = 0}, {x : f (x) < 1} ⊂ K,
Z
ϕ∞ ({x : f (x) < 1}) < ε,
| D f | dµ < ε.
For this purpose, assume A ≠ 0, denote the limit in the hypotheses of the lemma
by Q, and choose j ∈ P and xi ∈ A, 0 < ri < ∞ for i ∈ {1, . . . , j} with
µ B(xi , ri ) ≤ (Q + 1)rim and B(xi , ri ) ⊂ K
A⊂
j
[
i=1
ri
U xi ,
,
2
j
X
for i ∈ {1, . . . , j},
rim−1 <
i=1
ε
,
∆
where ∆ = sup{α(m − 1), 4(Q + 1)}. Selecting fi ∈ E (U, R) with 0 ≤ fi ≤ 1,
| D fi | ≤ 4ri−1 , and
2
whenever i ∈ {1, . . . , j}, one may take f =
{x : fi (x) < 1} ⊂ B(xi , ri )
Qj
i=1 fi
.
❐
ri
U xi ,
⊂ {x : fi (x) = 0},
Note 5.8 (see [Fed69, 1.7.5]).
Suppose
m, n ∈ P and m ≤ n. Then, one
V
V
defines the linear map γm : m Rn → m Rn by the equation
hξ, γm (η)i = ξ • η
whenever ξ, η ∈
^
m
Rn .
U LRICH M ENNE
1008
If we consider a subset E of a properly embedded submanifold M of Rn of
class 2, the distributional boundary of E with respect to M corresponds to variations of E tangential to M , and hence to the perimeter of E in M . In fact, allowing
for a small singular set of M , one obtains the following result that also applies to
varifolds corresponding to absolutely-area-minimising currents, as will be shown
in Remark 5.10.
Theorem 5.9. Suppose that m, n ∈ P , m ≤ n, U is an open subset of Rn ,
V ∈ IV m (U), 0 ≤ κ < ∞, kδV k ≤ κkV k, M is a relatively open subset of spt kV k,
H m−1 ((spt kV k) ∼ M) = 0, M is an m-dimensional submanifold of class 2,
τ : M → Hom(Rn , Rn ) satisfies τ(x) = Tan(M, x)♮ for x ∈ M , E is a kV k
measurable set, and
B = M ∼{x : Θ m (H
m
x M ∩ E, x) = 0 or Θ m (H m x M ∼ E, x) = 0}.
Then, the following three statements hold:
(1) If θ ∈ D (U, Rn ), then
Z
V ∂E(θ) = −
E∩M
Θ m (kV k, x) dH
τ(x) • Dhθ, τi(x)Θ
m
x.
(2) If ξ is an m vectorfield orienting M and γm is as in Note 5.8, then
m
V ∂E(θ) = (−1)
Z
E∩M
Θ m (kV k, x) dH
hξ(x), d(θ y(γm ◦ ξ))(x)iΘ
m
x
for θ ∈ D (U, Rn ).
(3) The distribution V ∂E is representable by integration if and only if
H m−1 (K ∩ B) < ∞
whenever K is a compact subset of U ;
in this case, B is H m−1 almost equal to {x : n(M, E ∩ M, x) ∈ Sn−1 }, meets
every compact subset of U in a (H m−1 , m − 1)-rectifiable set, and,
V ∂E(θ) = −
Z
M
Θ m (kV k, x) dH
n(M, E ∩ M, x) • θ(x)Θ
m−1
x
for θ ∈ D (U, Rn ).
Proof. Notice that the results of Allard [All72, 2.5, 4.5–4.7] stated for submanifolds of class ∞ have analogous formulations for submanifolds of class 2. The
meaning of Tan(M, θ), Nor(M, θ), Gm (M), and IV m (M) defined in [All72, 2.5,
3.1] will be extended accordingly.
The following assertion will be proven: if ξ is as in (2), and θ : U → Rn is a
vectorfield of class 1, then ϕ = γm ◦ ξ satisfies
hξ, d(θ y ϕ)(x)i = (−1)m−1 τ(x) • Dhθ, τi(x)
for x ∈ M.
Weakly Differentiable Functions on Varifolds
1009
For this purpose, assume that θ|M = Tan(M, θ). Noting |ξ(x)| = 1 for x ∈ M ,
one infers
hξ(x), hu, D ϕ(x)ii = ξ(x) • hu, D ξ(x)i = 0
for x ∈ M, u ∈ Tan(M, x).
Hence, for x ∈ M , one expresses ξ(x) = u1 ∧ · · · ∧ um for some orthonormal
basis u1 , . . . , um of Tan(M, x), and computes
hξ(x), d(θ y ϕ)(x)i
=
m
X
(−1)i−1 hu1 ∧ · · · ∧ ui−1 ∧ ui+1 ∧ · · · ∧ um , hui , D θ(x)i y ϕ(x)i
i=1
= (−1)m−1
m
X
hui , D θ(x)i • ui = (−1)m−1 τ(x) • D θ(x).
i=1
Next, the case M = spt kV k will be considered. In this case, one may assume
M is connected. Since
δV (θ) = 0
whenever θ ∈ D (U, Rn ) and Nor(M, θ) = 0
(e.g., by Brakke [Bra78, 5.8] or [Men13, 4.8]), Allard [All72, 4.6 (3)] then implies
for some 0 < λ < ∞ that
V (k) = λ
Z
M
k(x, Tan(M, x)) dH
Define W ∈ IV m (M) by W (k) =
that
V ∂E(θ) = −
Z
=−
Z
E
Z
m
E×G(n,m)
h(V , x) • θ(x) dkV kx −
E×G(n,m)
x
for k ∈ K (U × G(n, m)).
k dV for k ∈ K (Gm (M)), and notice
Z
E×G(n,m)
P♮ • D θ(x) dV (x, P )
P♮ • (D Tan(M, θ)(x) ◦ P♮ ) dV (x, P ) = −δW (Tan(M, θ))
whenever θ ∈ D (U, Rn ) by Allard [All72, 2.5 (2)]. Then, (1) is now evident,
and implies (2) by the assertion of the preceding paragraph. Concerning (3), the
subcase M = U follows from [Fed69, 4.5.6, 4.5.11] (recall also [Fed69, 4.1.28 (5),
4.2.1]), to which the case M = spt kV k may be reduced by applying Allard [All72,
4.5] to W (see also Allard [All72, 4.7]).
To treat the general case, first notice that, in view of Allard [All72, 5.1 (3)],
Lemma 5.7 is applicable with µ = kV k and A = (spt θ)∩(spt kV k) ∼ M whenever
θ ∈ D (U, Rn ), K is a compact subset of U , and spt θ ⊂ Int K . The resulting
functions fi ∈ E (U, R) satisfy
Z
|θ − fi θ| + | D θ − D(fi θ)| dkV k +
Z
X
|θ − fi θ| dH
m−1
→0
U LRICH M ENNE
1010
as i → ∞ whenever X is a H m−1 measurable subset of U with H m−1 (X) < ∞. It
follows that V ∂E(θ) = limi→∞ V ∂E(fi θ), and, if H m−1 (K ∩ B) < ∞, then
i→∞ M
Θ m (kV k, x) dH
n(M, E ∩ M, x) • (θ(x) − (fi θ)(x))Θ
Thus, one now readily verifies the assertion.
m−1
x = 0.
❐
lim
Z
loc (Rn ) is absolutely area minimising with respect
Remark 5.10. If S ∈ Rm
n
to R , ∂S = 0, and n − m = 1, then V ∈ IV m (Rn ) characterised by kSk = kV k
satisfies the hypotheses of Theorem 5.9 (2) with U = Rn , κ = 0 for some M and ξ
by Allard [All72, 4.8 (4)], [Fed69, 5.4.15], and Federer [Fed70, Theorem 1]; and
in this case, k∂(S x E)k = kV ∂Ek, as may be verified using Theorem 5.9 (2) (3) in
conjunction with [Fed69, 3.1.19, 4.1.14, 4.1.20, 4.1.30, 4.5.6].11
Remark 5.11. If one considers the situation m, n ∈ P , 1 < m < n, U is
an open subset of Rn , V ∈ IV m (U), and δV = 0, and few properties of V are
known to hold near H m−1 almost all x ∈ spt kV k. Consequently, it appears
difficult to obtain a structural description similar to Theorem 5.9 (3) for kV k
measurable sets whose distributional V boundary is representable by integration
in this more general situation. However, for L 1 almost all superlevel sets of a
real-valued generalised weakly differentiable function, such a description will be
proven under even milder hypotheses on V in Corollary 12.2.
6. D ECOMPOSITIONS OF VARIFOLDS
Introduction. In this section, the existence of a decomposition of rectifiable
varifolds whose first variation is representable by integration is established in Theorem 6.12. If the first variation is sufficiently well behaved, this decomposition
may be linked to the decomposition of the support of the weight measure into
connected components (see Corollary 6.14).
First, a useful set of hypotheses is gathered for later reference.
Hypothesis 6.1. Suppose m, n ∈ P , m ≤ n, 1 ≤ p ≤ m, U is an open subset of Rn , V ∈ Vm (U), kδV k is a Radon measure, and Θ m (kV k, x) ≥ 1 for kV k
n
almost all x . If p > 1, then suppose additionally that h(V , ·) ∈ Lloc
p (kV k, R )
and
Z
δV (θ) = − h(V , x) • θ(x) dkV kx for θ ∈ D (U, Rn ).
Therefore, V ∈ RV m (U) by Allard [All72, 5.5 (1)]. If Zp = 1, let ψ = kδV k. If
∗
p > 1 define a Radon measure ψ over U by ψ(A) =
|h(V , x)|p dkV kx for
A
A ⊂ U.
11With reference to [Fed69, 5.3.20] and Almgren [Alm00, 5.22], the hypothesis n − m = 1 could
have been omitted. However, the author has not checked Almgren’s result, and its consequences will
not be used in the present paper.
Weakly Differentiable Functions on Varifolds
1011
Definition 6.2. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn ,
V ∈ Vm (U), and kδV k is a Radon measure. Then, V is called indecomposable if
and only if there exists no kV k + kδV k measurable set E such that
kV k(E) > 0,
kV k(U ∼ E) > 0,
V ∂E = 0.
Remark 6.3. The same definition results if E is required to be a Borel set.
Remark 6.4. If V is indecomposable then so is λV for 0 < λ < ∞. This is in
contrast to a similar notion employed by Mondino in [Mon14, 2.15].
The following lemma shows a basic relation between the decomposability
properties of a varifold and the connectedness properties of the support of its
weight measure.
Lemma 6.5. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn , V ∈
Vm (U), kδV k is a Radon measure, E0 and E1 are nonempty, disjoint, relatively closed
subsets of spt kV k, and E0 ∪ E1 = spt kV k. Then, there holds
kV k(Ei ) > 0
and V ∂Ei = 0
for i ∈ {0, 1}. In particular, if V is indecomposable then spt kV k is connected.
Proof. Notice that U ∼ E0 and U ∼ E1 are open; hence,
kV k(E0 ) = kV k(U ∼ E1 ) > 0,
kV k(E1 ) = kV k(U ∼ E0 ) > 0.
To prove V ∂E1 (θ) = 0 for θ ∈ D (U, Rn ), choose w ∈ E (U, R) such that
for i ∈ {0, 1},
and apply Allard [All72, 4.10 (1)] with r , f , and g replaced by 21 , w , and θ.
❐
Ei ∩ spt θ ⊂ Int{x : w(x) = i}
Definition 6.6. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn ,
V ∈ Vm (U), and kδV k is a Radon measure. Then, W is called a component of V
if and only if 0 ≠ W ∈ Vm (U) is indecomposable and there exists a kV k + kδV k
measurable set E such that
W = V x E × G(n, m),
V ∂E = 0.
Remark 6.7. Suppose that F is a kV k + kδV k measurable set. Then, E is
kV k + kδV k almost equal to F if and only if W = V x F × G(n, m), and V ∂F = 0.
Remark 6.8. If C is a connected component of spt kV k and W is a component
of V with C ∩ spt kW k ≠ 0, then spt kW k ⊂ C by Lemma 6.5.
Definition 6.9. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn ,
V ∈ Vm (U), kδV k is a Radon measure, and Ξ ⊂ Vm (U). Then, Ξ is called a
decomposition of V if and only if the following three conditions are satisfied:
U LRICH M ENNE
1012
(1) Each member
of Ξ is a component of V .
P
(2) V (k) = W ∈Ξ
P W (k) whenever k ∈ K (U × G(n, m)).
(3) kδV k(f ) = W ∈Ξ kδW k(f ) for f ∈ K (U).
Remark 6.10. Clearly, Ξ is countable.
Moreover, there exists a function ξ mapping Ξ into the class of all Borel subsets
of U such that distinct members of Ξ are mapped onto disjoint sets, and
[
(kV k + kδV k) U ∼ im ξ = 0,
W = V x ξ(W ) × G(n, m) and V ∂ξ(W ) = 0
for W ∈ Ξ;
in fact, in view of Remark 6.7 it is sufficient to note that the last property implies
card{W : x ∈ ξ(W )} = 1 for kV k + kδV k almost all x.
S
Also, notice that V ∂( ξ[N]) = 0 whenever N ⊂ Ξ.
Remark 6.11. Suppose m, n, p, U , and V are as in Hypothesis 6.1, p = m,
and Ξ is a decomposition of V . Observe that Corollary 4.8 (5) if m = 1 and
[Men09, 2.5] if m > 1 imply that
card(Ξ ∩ {W : K ∩ spt kW k ≠ 0}) < ∞
whenever K is a compact subset of U ; hence,
spt kV k =
[
{spt kW k : W ∈ Ξ}.
Notice that [Men09, 1.2] readily shows that both assertions need not hold in the
case where p < m.
The main result of this section is the following theorem on the existence of a
decomposition for rectifiable varifolds.
Theorem 6.12. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn , V ∈
RV m (U), and kδV k is a Radon measure. Then, there exists a decomposition of V .
Proof. Assume V ≠ 0.
Denote by R the family of Borel subsets E of U such that V ∂E = 0. Notice
∞
\
Ei ∈ R
whenever Ei is a sequence in R with Ei+1 ⊂ Ei for i ∈ P ,
i=1
E∈R
if and only if E ∼ F ∈ R whenever E ⊃ F ∈ R
by Remark 5.3. Let P = R ∩ {E : kV k(E) > 0}. Next, define
δi = α(m)2−m−1 i−1−2m ,
εi = 2−1 i−2
Weakly Differentiable Functions on Varifolds
1013
for i ∈ P , and let Ai denote the Borel set of a ∈ Rn satisfying
|a| ≤ i,
U(a, 2εi ) ⊂ U,
kδV k B(a, r ) ≤ α(m)ir
m
Θ m (kV k, a) ≥
1
i
,
for 0 < r < εi
S
whenever i ∈ P . Clearly, Ai ⊂ Ai+1 for i ∈ P and kV k(U ∼ ∞
i=1 Ai ) = 0 by
Allard [All72, 3.5 (1a)] and [Fed69, 2.8.18, 2.9.5]. Moreover, define
Pi = R ∩ {E : kV k(E ∩ Ai ) > 0},
and notice that Pi ⊂ Pi+1 for i ∈ P and P =
bound given by
S∞
i=1 Pi
. One observes the lower
kV k(E ∩ B(a, εi )) ≥ δi
whenever E ∈ R , i ∈ P , a ∈ Ai , and Θ ∗m (kV k x E, a) ≥ 1/i; in fact, noting
Z εi
0
r −m kδ(V x E × G(n, m))k B(a, r ) dL 1 r ≤ α(m)iεi ,
the inequality follows from Corollary 4.5 and Remark 4.6. Let Qi denote the set
of E ∈ P such that there is no F satisfying
F ⊂ E,
F ∈ Pi ,
E ∼ F ∈ Pi .
Denote by Ω the class of Borel partitions H of U with H ⊂ P , and let also
G0 = {U} ∈ Ω. The previously observed lower bound implies
δi card(H ∩ Pi ) ≤ kV k(U ∩ {x : dist(x, Ai ) ≤ εi }) < ∞
whenever H is a disjointed subfamily of P and i ∈ P . In fact, for each E ∈ H ∩ Pi
there exists a ∈ Ai with Θ m (kV k x E, a) = Θ m (kV k, a) ≥ 1/i by [Fed69, 2.8.18,
2.9.11]; hence,
kV k(E ∩ {x : dist(x, Ai ) ≤ εi }) ≥ kV k(E ∩ B(a, εi )) ≥ δi .
In particular, such H is countable.
Next, one inductively (for i ∈ P ) defines Ωi to be the class of all H ∈ Ω such
that every E ∈ Gi−1 is the union of some subfamily of H , and chooses Gi ∈ Ωi
such that
card(Gi ∩ Pi ) ≥ card(H ∩ Pi ) whenever H ∈ Ωi .
The maximality of Gi implies Gi ⊂ Qi ; in fact, if there existed E ∈ Gi ∼ Qi there
would exist F satisfying
F ⊂ E,
F ∈ Pi ,
E ∼ F ∈ Pi ,
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and H = (Gi ∼{E}) ∪ {F , E ∼ F } would belong to Ωi with
card(H ∩ Pi ) > card(Gi ∩ Pi ).
Moreover, it is evident that to each x ∈ U there
T corresponds a sequence Ei
uniquely characterised by the requirements x ∈ ∞
i=1 Ei and Ei+1 ⊂ Ei ∈ Gi
for i ∈ P .
S∞
Define G T
= i=1 Gi , and notice that G is countable. Define C to be the collection of sets ∞
i=1 Ei with positive kV k measure corresponding to all sequences Ei
with Ei+1 ⊂ Ei ∈ Gi for i ∈ P . Clearly, C is a disjointed subfamily of P , and
hence C is countable. Next, it will be shown that
[ kV k U ∼ C = 0.
In view of [Fed69, 2.8.18, 2.9.11] it is sufficient to prove
Ai ∼
[
C⊂
[
E ∩ {x : Θ ∗m (kV k x E, x) < Θ ∗m (kV k, x)} : E ∈ G
for i ∈ P . For this purpose, consider a ∈ Ai ∼
quence Ej . It follows that
kV k
∞
\
j=1
Ej = 0.
S
C with corresponding se-
Hence, there exists j with kV k(Ej ∩ B(a, εi )) < δi , and the lower bound implies
Θ ∗m (kV k x Ej , a) <
1
i
≤ Θ m (kV k, a).
It remains to prove that each varifold V x E ×G(n, m) correspondingTto E ∈ C
is indecomposable. If this were not the case, then there would exist E = i=1 Ei ∈
C with Ei+1 ⊂ Ei ∈ Gi for i ∈ P and a Borel set F such that
kV k(E ∩ F ) > 0,
kV k(E ∼ F ) > 0,
V ∂(E ∩ F ) = 0
by Remarks 5.4 and 6.3. This would imply E ∩ F ∈ P and E ∼ F ∈ P ; hence, for
some i also E ∩ F ∈ Pi and E ∼ F ∈ Pi , which would yield
E ∩ F ⊂ Ei ,
Ei ∼(E ∩ F ) ∈ Pi ,
❐
since E ∼ F ⊂ Ei ∼(E ∩ F ) ∈ R , which is a contradiction to Ei ∈ Qi .
Remark 6.13. The decomposition of V may be nonunique. In fact, considering the six rays
π ij
Rj = t exp
: 0 < t < ∞ ⊂ C = R2 ,
3
where π = Γ
2
1
2
,
Weakly Differentiable Functions on Varifolds
1015
corresponding to j ∈ {0, 1, 2, 3, 4, 5} and their associated varifolds Vj ∈ IV 1 (R2 )
P
with kVj k = H 1 x Rj , one notices that V = 5j=0 Vj ∈ IV 1 (R2 ) is a stationary
varifold such that
{V0 + V2 + V4 , V1 + V3 + V5 }
and
{V0 + V3 , V1 + V4 , V2 + V5 }
are distinct decompositions of V .
With the previous theorem at one’s disposal, the connectedness structure of
the support of the weight measure in case p = m is now readily understood.
Corollary 6.14. Suppose m, n, p, U and V are as in Hypothesis 6.1, p = m,
and Φ is the family of all connected components of spt kV k. Then, the following four
statements hold:
S
(1) If C ∈ Φ, then C = {spt kW k : W ∈ Ξ, C ∩ spt kW k ≠ 0} whenever Ξ is
a decomposition of V .
(2) card(Φ ∩ {C : C ∩ K ≠ 0}) < ∞ whenever K is a compact subset of U .
(3) If C ∈ Φ, then C is open relative to spt kV k.
(4) If C ∈ Φ, then spt(kV k x C) = C and V ∂C = 0.
❐
Proof. First, (1) is a consequence of Remarks 6.8 and 6.11. In view of (1)
and Theorem 6.12, (2) is a consequence of Remark 6.11. Next, (2) implies (3).
Finally, (3) and Lemma 6.5 yield (4).
Remark 6.15. If V is stationary, then (4) implies that V x C × G(n, m) is
stationary. This fact might prove useful in considerations involving a strong maximum principle, such as Wickramasekera [Wic14, Theorem 1.1].
7. R ELATIVE I SOPERIMETRIC I NEQUALITY
Introduction. In this section, a general isoperimetric inequality for varifolds
satisfying a lower density bound is established (see Theorem 7.8). As corollaries,
one obtains two relative isoperimetric inequalities under the relevant conditions
on the first variation of the varifold (see Corollaries 7.9 and 7.11).
First, a “zero boundary value” condition for sets on varifolds is introduced.
Hypothesis 7.1. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn ,
V ∈ Vm (U), kδV k is a Radon measure, Θ m (kV k, x) ≥ 1 for kV k almost all x ,
E is kV k + kδV k measurable, B is a closed subset of Bdry U , and (see Definitions
2.18 and 2.20 and Remark 2.19),
Z
(kV k + kδV k)(E ∩ K) + kV ∂Ek(U ∩ K) < ∞,
E×G(n,m)
P♮ • D θ(x) dV (x, P ) = ((δV ) x E)(θ|U) − (V ∂E)(θ|U)
whenever K is a compact subset of Rn ∼ B and θ ∈ D (Rn ∼ B, Rn ). Defining
W ∈ Vm (Rn ∼ B) by
W (A) = V (A ∩ (E × G(n, m)))
for A ⊂ (Rn ∼ B) × G(n, m),
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1016
we see this implies
kδW k(A) ≤ kδV k(E ∩ A) + kV ∂Ek(U ∩ A)
for A ⊂ Rn ∼ B.
Example 7.2. Using Theorem 5.9 (1) (3), one verifies the following statement. If m, n ∈ P , m ≤ n, U is an open subset of Rn , B is a closed subset
of Bdry U , M is an m-dimensional submanifold of Rn of class 2, M ⊂ Rn ∼ B ,
(Clos M) ∼ M ⊂ B , V ∈ Vm (U) and V ′ ∈ Vm (Rn ∼ B) satisfy
V (k) =
V ′ (k) =
Z
Z
M∩U
M
k(x, Tan(M, x)) dH
k(x, Tan(M, x)) dH
m
m
x
x
for k ∈ K (U × G(n, m)),
for k ∈ K ((Rn ∼ B) × G(n, m)),
and E is an H m measurable subset of M ∩U , then E satisfies the conditions of Hypothesis 7.1 if and only if V ′ ∂E is representable by integration and n(M, E, x) = 0
for H m−1 almost all x ∈ M ∼ U ; in this case,
V ∂E(θ|U) = −
Z
M
n(M, E, x) • θ(x) dH
m−1
for θ ∈ D (Rn ∼ B, Rn ).
x
Since, in general, in the situation of Hypothesis 7.1 there is no varifold V ′ available
that extends V in a canonical way, the condition on E is formulated in terms of
the behaviour of W .
The results of the present section rest on the following lemma obtained by a
simple compactness argument.
Lemma 7.3. Suppose 1 ≤ M < ∞. Then, there exists a positive, finite number
Γ with the following property. If m, n ∈ P , m ≤ n ≤ M , 1 ≤ Q ≤ M , a ∈ Rn ,
0 < r < ∞, W ∈ Vm (U(a, r )), kδW k is a Radon measure, Θ m (kW k, x) ≥ 1 for
kW k almost all x , a ∈ spt kW k, and
kδW k B(a, s) ≤ Γ −1 kW k(B(a, s))1−1/m
m
then
kW k({x : Θ (kW k, x) < Q}) ≤ Γ
−1
for 0 < s < r ,
kW k U(a, r ),
kW k U(a, r ) ≥ (Q − M −1 )α(m)r m .
Proof. If the lemma were false for some M , there would exist a sequence Γi
with Γi → ∞ as i → ∞, and sequences mi , ni , Qi , ai , ri , and Wi showing that
Γ = Γi does not have the asserted property.
One could assume for some m, n ∈ P , 1 ≤ Q ≤ M that m ≤ n ≤ M ,
m = mi , n = ni , ai = 0, and ri = 1 for i ∈ P and Qi → Q as i → ∞, and, by
[Men09, 2.5],
kWi k B(0, s) ≥ (2mγ (m))−m s m
whenever 0 < s < 1 and i ∈ P .
Weakly Differentiable Functions on Varifolds
1017
Defining W ∈ Vm (Rn ∩ U(0, 1)) to be the limit of some subsequence of Wi , one
would obtain
kW k U(0, 1) ≤ (Q − M −1 )α(m),
0 ∈ spt kW k,
δW = 0.
Finally, using Allard [All72, 5.1 (2), 5.4, 8.6], one would then conclude that
Θ m (kW k, x) ≥ Q
for kW k almost all x,
Θ (kW k, 0) ≥ Q, kW k U(0, 1) ≥ Qα(m),
a contradiction.
❐
m
Remark 7.4. Considering stationary varifolds whose support is contained in
two affine planes with Θ m (kW k, a) a small positive number shows that the hypothesis “Θ m (kW k, x) ≥ 1 for kW k almost all x ” cannot be omitted even for
rectifiable varifolds.
Remark 7.5. Even for functions of class 1, Lemma 7.3 is the key observation
that—through the relative isoperimetric inequalities Corollaries 7.9 and 7.11—
leads to Sobolev Poincaré-type estimates that are applicable near kV k almost all
points of {x : Θ m (kV k, x) ≥ 2} (see Theorem 10.1). For generalised weakly
differentiable functions, these estimates in turn provide an important ingredient
for the differentiability results obtained in Theorems 11.2 and 11.4 and the coarea
formula in Corollary 12.2.
Remark 7.6. Taking Q = 1 in Lemma 7.3 (or applying [Men09, 2.6]) yields
the following proposition: if m, n, p, U , V , and ψ are as in Hypothesis 6.1,
p = m, a ∈ spt kV k, and ψ({a}) = 0, then Θ m
∗ (kV k, a) ≥ 1. If kδV k is
absolutely continuous with respect to kV k then the condition ψ({a}) = 0 is
redundant.
If m = n and f : Rn → {y : 1 ≤ y < ∞} is a weakly differentiable function
n
n
n
with D f ∈ Lloc
n (L , Hom(R
Z , R)), then the varifold V ∈ RV n (R ) defined by
the requirement kV k(B) =
B
f dL n whenever B is a Borel subset of Rn satisfies
the conditions of Hypothesis 6.1 with p = n since
Z
δV (θ) = − hθ(x), D(log ◦f )(x)i dkV kx
for θ ∈ D (Rn , Rn );
here, the terminology and notation for weakly differentiable functions of [Men12a,
pp. 9–10] is employed. If n > 1 and C is a countable subset of Rn , one may use
well-known properties of Sobolev functions (in particular, example [AF03, 4.43])
to construct f such that C ⊂ {x : Θ n (kV k, x) = ∞}. It is therefore evident that
the conditions of Hypothesis 6.1 with p = m > 1 are insufficient to guarantee
finiteness or upper semicontinuity of Θ m (kV k, ·) at each point of U . However,
the following proposition was obtained by Kuwert and Schätzle in [KS04, Appendix A]: if m, n, p, U , and V are as in Hypothesis 6.1, p = m = 2, and
U LRICH M ENNE
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V ∈ IV 2 (U), then Θ 2 (kV k, ·) is a real-valued, upper semicontinuous function
whose domain is U .
Remark 7.7. The preceding remark is a corrected and extended version of
the author’s remark in [Men09, 2.7], where the last two sentences should have
referred to integral varifolds.
The main theorem of this section is the following general isoperimetric inequality that allows us to unify the treatment of its two corollaries.
Theorem 7.8. Suppose 1 ≤ M < ∞. Then, there exists a positive, finite number Γ
with the following property. If m, n ∈ P , m ≤ n ≤ M , 1 ≤ Q ≤ M , U is an
open subset of Rn , W ∈ Vm (U), S, Σ ∈ D ′ (U, Rn ) are representable by integration,
δW = S + Σ, Θ m (kW k, x) ≥ 1 for kW k almost all x , 0 < r < ∞, and
kSk(U) ≤ Γ −1
S(θ) ≤ Γ
−1
if m = 1,
kW k(m/(m−1)) (θ)
for θ ∈ D (U, Rn ) if m > 1,
kW k(U) ≤ (Q − M −1 )α(m)r m ,
then
kW k({x : Θ m (kW k, x) < Q}) ≤ Γ −1 r m ,
where 00 = 0.
kW k({x : U(x, r ) ⊂ U})1−1/m ≤ Γ kΣk(U),
Proof. Define
∆1 = Γ7.3 (2M)−1 ,
∆2 = inf{(2γ (m))−1 : M ≥ m ∈ P },
∆3 = ∆1 inf{(2mγ (m))−m : M ≥ m ∈ P },
∆5 =
1
inf{∆2 , ∆3 },
2
∆4 = sup{β (n) : M ≥ n ∈ P },
1
Γ = ∆4 ∆−
5 .
Notice that γ (1) ≥ 12 , and hence 2∆5 ≤ Γ7.3 (2M)−1 .
Suppose m, n, Q, U , W , S , Σ, and r satisfy the hypotheses in the body of the
theorem with Γ .
Abbreviate A = {x : U(x, r ) ⊂ U}. Clearly, if m = 1 then kSk(U) ≤ ∆5 .
Observe that, if m > 1, then kSk(X) ≤ ∆5 kW k(X)1−1/m whenever X ⊂ U .
Next, the following assertion will be shown: if a ∈ A ∩ spt kW k, then there
exists 0 < s < r such that
∆5 kW k(B(a, s))1−1/m < kΣk B(a, s).
Since kδW k B(a, s) ≤ ∆5 kW k(B(a, s))1−1/m + kΣk B(a, s), it is sufficient to exhibit 0 < s < r with
2∆5 kW k(B(a, s))1−1/m < kδW k B(a, s).
Weakly Differentiable Functions on Varifolds
1019
As kW k U(a, r ) ≤ kW k(U) < (Q−(2M)−1 )α(m)r m , the nonexistence of such s
would imply by use of [Men09, 2.5] and Lemma 7.3 that
∆1 (2mγ (m))−m r m ≤ ∆1 kW k U(a, r )
< kW k(U(a, r ) ∩ {x : Θ m (kW k, x) < Q})
≤ ∆5 r m ,
a contradiction.
By the assertion of the preceding paragraph, there exist countable disjointed
families of closed balls G1 , . . . , Gβ(n) such that
A ∩ spt kW k ⊂
[[
{Gi : i = 1, . . . , β (n)} ⊂ U,
1
kW k(B)1−1/m ≤ ∆−
5 kΣk(B)
whenever B ∈ Gi and i ∈ {1, . . . , β (n)}.
If m > 1, then, defining β = m/(m − 1), one estimates
kW k(A) ≤
βX
(n)
X
i=1 B∈Gi
−β
≤ ∆5
βX
(n) i=1
−β
kW k(B) ≤ ∆5
X
B∈Gi
βX
(n)
X
kΣk(B)β
i=1 B∈Gi
β
−β
kΣk(B) ≤ ∆5 β (n)kΣk(U)β .
❐
If mS= 1 and kW k(A) > 0, then we have ∆5 ≤ kΣk(B) ≤ kΣk(U) for some
B ∈ {Gi : i = 1, . . . , β (n)}.
The relative isoperimetric inequality involving the distributional boundary of
a set is a direct consequence of the preceding theorem.
Corollary 7.9. Suppose m, n, U , V , E , and B are as in Hypothesis 7.1, n ≤
M < ∞, 1 ≤ Q ≤ M , Λ = Γ7.8 (M), 0 < r < ∞, and
kV k(E) ≤ (Q − M −1 )α(m)r m ,
Then,
kV k(E ∩ {x : Θ m (kV k, x) < Q}) ≤ Λ−1 r m .
kV k(E ∩ {x : U(x, r ) ⊂ Rn ∼ B})1−1/m ≤ Λ(kV ∂Ek(U) + kδV k(E)),
where 00 = 0.
Proof. Define W as in Hypothesis 7.1, and note that
Θ m (kW k, x) = Θ m (kV k, x) ≥ 1
❐
for kW k almost all x by [Fed69, 2.8.18, 2.9.11]. Hence, applying Theorem 7.8
with U , S , and Σ replaced by Rn ∼ B , 0, and δW yields the conclusion.
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Remark 7.10. The case B = Bdry U and Q = 1 corresponds to Hutchinson
[Hut90, Theorem 1], which is formulated in the context of functions rather than
sets.
If p = m in Hypothesis 6.1 and a smallness condition on ψ (see Hypothesis
6.1) is satisfied, then the previous inequality may be sharpened by omission of the
term involving δV .
Corollary 7.11. Suppose m, n, p, U , V , and ψ are as in Hypothesis 6.1,
p = m, E and B are related to m, n, U , and V as in Hypothesis 7.1, n ≤ M < ∞,
1 ≤ Q ≤ M , Λ = Γ7.8 (M), 0 < r < ∞, and
kV k(E) ≤ (Q − M −1 )α(m)r m ,
ψ(E)1/m ≤ Λ−1 ,
kV k(E ∩ {x : Θ m (kV k, x) < Q}) ≤ Λ−1 r m .
Then,
kV k(E ∩ {x : U(x, r ) ⊂ Rn ∼ B})1−1/m ≤ ΛkV ∂Ek(U),
where 00 = 0.
Proof. Define W as in Hypothesis 7.1, and note also that Θ m (kW k, x) =
Θ (kV k, x) ≥ 1 for kW k almost all x by [Fed69, 2.8.18, 2.9.11]. If m > 1,
then approximation shows
m
((δV ) x E)(θ|U) = −
Z
E
h(V , x) • θ(x) dkV kx
for θ ∈ D (Rn ∼ B, Rn );
❐
in fact, since k(δV ) x Ek = kδV k x E and θ|U ∈ L1 (kδV k x E, Rn ), the problem reduces to the case spt θ ⊂ U , which is readily treated. Therefore, applying
Theorem 7.8 with U , S(θ), and Σ(θ) replaced by Rn ∼ B , ((δV ) x E)(θ|U), and
−(V ∂E)(θ|U) yields the conclusion.
Remark 7.12. Evidently, considering E = U , B = Bdry U , Q = 1, and stationary varifolds V such that kV k(U) is a small positive number shows that the
intersection with {x : U(x, r ) ⊂ Rn ∼ B} cannot be omitted in the conclusion.
Remark 7.13. Since estimating kδV k(E) by means of the Hölder’s inequality seems insufficient to derive Corollary 7.11 from Corollary 7.9 by means of
“absorption,” this procedure was implemented at an earlier stage and led to the
formulation of Theorem 7.8.
Remark 7.14. It is instructive to consider the following situation. Suppose
that m, n ∈ P , m ≤ n, 1 ≤ M < ∞, a ∈ Rn , 0 < r < ∞, 0 ≤ κ < ∞,
1 < λ < ∞, U = U(a, λr ), B = Bdry U , A = {x : U(x, r ) ⊂ Rn ∼ B} (hence,
A = B(a, (λ − 1)r )), V ∈ IV m (U) is a stationary varifold, and E is a kV k measurable set satisfying the relative isoperimetric estimate
inf{kV k(A ∩ E), kV k(A ∼ E)}1−1/m ≤ κkV ∂Ek(U),
Weakly Differentiable Functions on Varifolds
1021
where 00 = 0. Then, kV k(A ∩ E) ≤ (1 − M −1)kV k(A) implies
kV k(A ∩ E)1−1/m ≤ MκkV ∂Ek(U).
Exhibiting a suitable class of V and E such that the relative isoperimetric estimate holds with a uniform number κ is complicated by the absence of such an
estimate on the catenoid.12 If V corresponds to an absolutely-area-minimising
locally rectifiable current in codimension one, then such uniform control was obtained for some λ in Bombieri and Giusti [BG72, Theorem 2] (and attributed by
these authors to De Giorgi); see Remark 5.10 for the link of the concept of distributional boundary employed by Bombieri and Giusti and the one of the present
paper.
Finally, the term “relative isoperimetric inequality” (or estimate) is chosen in
accordance with the usage of that term in the case kV k = L n by Ambrosio, Fusco,
and Pallara (see [AFP00, (3.43), p. 152]).
8. G ENERALISED W EAKLY D IFFERENTIABLE F UNCTIONS
Introduction. In this section, generalised weakly differentiable functions are
introduced in Definition 8.3. Properties studied include behaviour under composition (see Lemma 8.12, Example 8.13 and Lemma 8.15), addition and multiplication (see Theorem 8.20 (3) (4)), and decomposition of the varifold (see Theorems 8.24 and 8.34). Moreover, coarea formulae in terms of the distributional
boundary of superlevel sets are established (see Lemma 8.1 and Theorem 8.30).
A measure-theoretic description of the boundary of the superlevel sets will appear
in Corollary 12.2. The theory is illustrated by examples in Example 8.25 and
Example 8.28.
First, it will be observed that, for every measurable real-valued function, there
exists a unique distribution (related to the “derivative” of the function; see Remark
8.5) that satisfies certain integration-by-parts identities. Additionally, an abstract
coarea-type formula may be phrased in terms of this distribution and the superlevel
sets of the function.
Lemma 8.1. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn , V ∈
Vm (U), kδV k is a Radon measure, f is a real-valued kV k + kδV k measurable
function with dmn f ⊂ U , and E(y) = {x : f (x) > y} for y ∈ R.
Then, there exists a unique T ∈ D ′ (U × R, Rn ) such that (see Remark 2.19)
(δV )((ω ◦ f )θ) =
Z
ω(f (x))P♮ • D θ(x) dV (x, P ) + T(x,y) (ω′ (y)θ(x))
whenever θ ∈ D (U, Rn ), ω ∈ E (R, R), spt ω′ is compact and inf spt ω > −∞.
Moreover,
T (ϕ) =
Z
V ∂E(y)(ϕ(·, y)) dL 1 y
for ϕ ∈ D (U × R, Rn ).
12See, e.g., [Oss86, p. 18] for a description of the catenoid.
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Proof. Define C = {(x, y) : x ∈ E(y)} and g : (U × G(n, m)) × R →
U × R by g((x, P ), y) = (x, y) for x ∈ U , P ∈ G(n, m) and y ∈ R. Using
[Fed69, 2.2.2, 2.2.3, 2.2.17, 2.6.2], one obtains that C is kV k × L 1 measurable
and kδV k × L 1 measurable, and hence that g −1 [C] is V × L 1 measurable since
kV k × L 1 = g# (V × L 1 ).
Define p : Rn × R → Rn by p(x, y) = x for (x, y) ∈ Rn × R, and let
T ∈ D ′ (U × R, Rn ) be defined by
Z
T (ϕ) =
η(V , x) • ϕ(x, y) d(kδV k × L 1 )(x, y)
C
Z
−
P♮ • (D ϕ(x, y) ◦ p ∗ ) d(V × L 1 )((x, P ), y)
g −1 [C]
❐
whenever ϕ ∈ D (U × R, Rn ) (see Hypothesis 4.4). Fubini’s theorem then yields
the two equations. The uniqueness of T follows from Lemma 3.1.
Remark 8.2. Notice that the characterising equation for T also holds if the
requirement inf spt ω > −∞ is dropped.
Next, the concept of generalised weakly differentiable function is defined. In
order to single out a “canonical representative” for the generalised weak derivative,
the concept of approximate limit is employed (see [Fed69, 2.9.12]).
Definition 8.3. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn ,
V ∈ RV m (U), kδV k is a Radon measure, and Y is a finite-dimensional normed
vector space.
Then, a Y -valued kV k + kδV k measurable function f with dmn f ⊂ U is
called generalised V weakly differentiable if and only if, for some kV k measurable
Hom(Rn , Y )-valued function F , the following two conditions hold:
(1) If K is a compact subset of U and 0 ≤ s < ∞, then
Z
K∩{x : |f (x)|≤s}
kF k dkV k < ∞.
(2) If θ ∈ D (U, Rn ), γ ∈ E (Y , R) and spt D γ is compact, then (see Remark
2.19)
(δV )((γ ◦ f )θ) =
Z
γ(f (x))P♮ • D θ(x) dV (x, P )
Z
+ hθ(x), D γ(f (x)) ◦ F (x)i dkV kx.
The function F is kV k almost unique. Therefore, one may define the generalised
V weak derivative of f to be the function V D f characterised by a ∈ dmn V D f
if and only if
(kV k, C) ap lim F (x) = σ
x→a
for some σ ∈ Hom(Rn , Y ),
Weakly Differentiable Functions on Varifolds
1023
and, in this case, V D f (a) = σ , where C = {(a, B(a, r )) : B(a, r ) ⊂ U}. Moreover, the set of all Y -valued generalised V weakly differentiable functions will be
denoted by T(V , Y ) and T(V ) = T(V , R).
Remark 8.4. Condition (2) is equivalent to the following condition:
(2)′ If ζ ∈ D (U, R), u ∈ Rn , γ ∈ E (Y , R) and spt D γ is compact, then
(δV )((γ ◦ f )ζ · u) =
Z
γ(f (x))hP♮ (u), D ζ(x)i
+ ζ(x)hu, D γ(f (x)) ◦ F (x)i dV (x, P ).
If dim Y ≥ 2 or f is locally bounded, then one may require spt γ to be compact
in (2) or (2)′ , as dim Y ≥ 2 implies Y ∼ B(0, s) is connected for 0 < s < ∞.
Remark 8.5. If f ∈ T(V ), then the distribution T associated with f in
Lemma 8.1 is representable by integration, and (see Remark 2.15, Lemma 3.1,
and Lemma 3.3 with J = R),
T (ϕ) =
Z
g dkT k =
Z
Z
hϕ(x, f (x)), V D f (x)i dkV kx,
g(x, f (x))|V D f (x)| dkV kx
whenever ϕ ∈ L1 (kT k, Rn) and g is an R̄-valued kT k integrable function.
Remark 8.6. If f ∈ T(V , Y ), θ : U → Rn is Lipschitzian with compact
support, γ : Y → R is of class 1, and either spt D γ is compact or f is locally
bounded, then
(δV )((γ ◦ f )θ) =
Z
γ(f (x))P♮ • ((kV k, m) ap D θ(x) ◦ P♮ ) dV (x, P )
Z
+ hθ(x), D γ(f (x)) ◦ V D f (x)i dkV kx,
as may be verified by means of approximation and [Men12a, 4.5 (3)]. Consequently, if f ∈ T(V , Y ) is locally bounded, Z is a finite-dimensional normed
vector space, and g : Y → Z is of class 1, then g ◦ f ∈ T(V , Z) with
V D (g ◦ f )(x) = D g(f (x)) ◦ V D f (x)
for kV k almost all x.
Example 8.7. If f : U → Y is a locally Lipschitzian function, then f is
generalised V weakly differentiable with
m
V D f (x) = (kV k, m) ap D f (x) ◦ Tan (kV k, x)♮
for kV k almost all x,
as may be verified by means of [Men12a, 4.5 (4)]. Moreover, if Θ m (kV k, x) ≥ 1
for kV k almost all x , then the equality holds for any f ∈ T(V , Y ), as will be
shown in Theorem 11.2.
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Remark 8.8. The prefix “generalised” has been chosen in analogy with the
notion of “generalised function of bounded variation” treated in [AFP00, Section
4.5] originating from De Giorgi and Ambrosio [DGA88].
Remark 8.9. The usefulness of integration-by-parts identities involving the
first variation in defining a concept of weakly differentiable functions on varifolds
has already been “expected” by Anzellotti, Delladio, and Scianna who developed
two notions of functions of bounded variation on integral currents (see [ADS96,
p. 261]).
Remark 8.10. In order to define a concept of “generalised (real-valued) function of locally bounded variation” with respect to a varifold, it could be of interest
to study the class of those functions f satisfying the hypotheses of Lemma 8.1
such that the associated function T is representable by integration.
Remark 8.11. A concept related to the present one has been proposed by
Moser in [Mos01, Definition 4.1] in the context of curvature varifolds (see Definition 15.4 and Remark 15.5); in fact, it allows for certain “multiple-valued”
functions. In studying the convergence of pairs of varifolds and weakly differentiable functions, it would seem natural to investigate the extension of the present
concept to such functions. (Notice that the usage of the term “multiple-valued”
here is different, but related to the one of Almgren in [Alm00, Section 1]).
The definition of a generalised weakly differentiable function directly entails
a basic stability result for compositions with suitably smooth functions.
Lemma 8.12. Suppose that m, n, U , V , and Y are as in Definition 8.3,
f ∈ T(V , Y ), Z is a finite-dimensional normed vector space, 0 ≤ κ < ∞, Υ is a
closed subset of Y , g : Y → Z , H : Y → Hom(Y , Z), gi : Y → Z is a sequence of
functions of class 1, spt D gi ⊂ Υ , Lip gi ≤ κ , g|Υ is proper, and
g(y) = lim gi (y)
uniformly in y ∈ Y ,
H(y) = lim D gi (y)
for y ∈ Y .
i→∞
i→∞
Then, g ◦ f ∈ T(V , Z) and
V D (g ◦ f )(x) = H(f (x)) ◦ V D f (x)
for kV k almost all x.
Proof. Note Lip g ≤ κ , kH(y)k ≤ κ for y ∈ Y , and H|Y ∼ Υ = 0; hence,
{y : |g(y)| ≤ s} ⊂ {y : H(y) = 0} ∪ (g|Υ)−1 [B(0, s)],
Z
kH(f (x)) ◦ V D f (x)k dkV kx < ∞
K∩{x : |g(f (x))|≤s}
whenever K is a compact subset of U and 0 ≤ s < ∞.
Suppose γ ∈ E (Z, R) and 0 ≤ s < ∞ with spt D γ ⊂ U(0, s) and C =
(g|Υ)−1 [B(0, s)]. Then, im γ is bounded, C is compact, and
Y ∩ {y : D γ(gi (y)) ◦ D gi (y) ≠ 0} ⊂ (gi |Υ)−1 [spt D γ] ⊂ C
for large i;
Weakly Differentiable Functions on Varifolds
1025
in particular, spt D(γ ◦ gi ) ⊂ C for such i. By using Remark 8.6, it follows that
for θ ∈ D (U, Rn ); then, considering the limit i → ∞ yields the conclusion.
❐
Z
(δV )((γ ◦ gi ◦ f )θ) = γ(gi (f (x)))P♮ • D θ(x) dV (x, P )
Z
+ hθ(x), D γ(gi (f (x))) ◦ D gi (f (x)) ◦ V D f (x)i dkV kx
Example 8.13. Amongst the functions g and H admitting an approximation
as in Lemma 8.12 are the following:
(1) If L : Y → Y is a linear automorphism of Y , then g = L and H = D L is
admissible.
(2) If b ∈ Y , then g = τb with H = D τb is admissible.
(3) If Y is an inner product space and b ∈ Y , then one may take g and H
such that, g(y) = |y − b| for y ∈ Y ,
H(y)(v) = |y − b|−1 (y − b) • v if y ≠ b,
H(y) = 0 if y = b
whenever v, y ∈ Y .
(4) If Y = R and b ∈ R, then one may take g and H such that
g(y) = sup{y, b},
H(y)(v) = v if y > b,
H(y) = 0 if y ≤ b
whenever v, y ∈ R.
Here, (1) and (2) are trivial.
To prove (3), assume Y =Z Rℓ for some ℓ ∈ P by (1), choose a nonnegative
function ̺ ∈ D (Y , R) with ̺ dL ℓ = 1 and ̺(y) = ̺(−y) for y ∈ Y , and
take κ = 1, Υ = Y , and gi = ̺1/i ∗ g in Lemma 8.12, noting (̺1/i ∗ g)(b − y) =
(̺1/i ∗ g)(b + y) for y ∈ Y ; hence, D(̺1/i ∗ g)(b) = 0.
Z
To prove (4), choose a nonnegative function ̺ ∈ D (R, R) with ̺ dL 1 = 1,
spt ̺ ⊂ B(0, 1), and ε = inf spt ̺ > 0, and take κ = 1, Υ = R ∩ {y : y ≥ b}, and
gi = ̺1/i ∗ g in Lemma 8.12, noting gi (y) = b if −∞ < y ≤ b + ε/i; hence,
D gi (y) = 0 for −∞ < y ≤ b.
To prepare for a more general composition result, certain properties of the
weak topology on L1 (µ, Y ) will be collected. Recall there is no identification of
functions agreeing µ almost everywhere in L1 (µ, Y ).
Lemma 8.14. Suppose U is an open subset of Rn , µ is a Radon measure over U ,
Y is a finite-dimensional normed vector space, g ∈ L1 (µ), and K denotes the set of all
f ∈ L1 (µ, Y ) such that
|f (x)| ≤ g(x)
for µ almost all x,
U LRICH M ENNE
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L1 (µ, Y ) = L1 (µ, Y )/{f : µ(1) (f ) = 0} is the (usual ) quotient Banach space, and
π : L1 (µ, Y ) → L1 (µ, Y ) denotes the canonical projection.
Then, π [K] with the topology induced by the weak topology on L1 (µ, Y ) is com-
pact and metrisable.
❐
Proof. First, notice that π [K] is convex and closed, and hence weakly closed
by [DS58, 5.3.13]. Therefore, one may assume that Y = R as a basis of Y induces a linear homeomorphism L1 (µ, R)dim Y ≃ L1 (µ, Y ) with respect to the weak
topologies on L1 (µ, R) and L1 (µ, Y ). Since L1 (µ, R) is separable, the conclusion
now follows combining [DS58, 4.8.9, 5.6.1, 5.6.3].
Now, compositions with suitably proper Lipschitzian function may be treated.
Lemma 8.15. Suppose m, n, U , V , and Y are as in Definition 8.3, f ∈
T(V , Y ), Z is a finite-dimensional normed vector space, Υ is a closed subset of Y , c is
the characteristic function of f −1[Υ], and g : Y → Z is a Lipschitzian function such
that g|Υ is proper and g|Y ∼ Υ is locally constant.
Then, g ◦ f ∈ T(V , Z) and
kV D (g ◦ f )(x)k ≤ Lip(g)c(x)kV D f (x)k
for kV k almost all x.
Proof. Suppose 0 < ε ≤ 1, and abbreviate κ = Lip g .
Define B = Y ∩ {y : dist(y, Υ) ≤ ε}, and let b denote the characteristic function of f −1[B]. Since g|B is proper, one may employ convolution to construct
gi ∈ E (Y , Z) satisfying Lip gi ≤ κ , spt D gi ⊂ B , and
δi = sup{|(g − gi )(y)| : y ∈ Y } → 0
as i → ∞.
Therefore, if δi < ∞, then gi |B is proper and gi ◦ f ∈ T(V , Z) with
kV D (gi ◦ f )(x)k ≤ κb(x)kV D f (x)k
for kV k almost all x
by Lemma 8.12 with Υ , g , and H replaced by B , gi , and D gi . Choose
S a sequence
of compact sets Kj such that Kj ⊂ Int Kj+1 for j ∈ P and U = ∞
j=1 Kj , and
define E(j) = Kj ∩ {x : |f (x)| < j} for j ∈ P . In view of Lemma 8.14, possibly
passing to a subsequence by means of a diagonal process, there exist functions
Fj ∈ L1 (kV k x E(j), Hom(Rn , Z)) such that
kFj (x)k ≤ κb(x)kV D f (x)k for kV k almost all x ∈ Ej ,
Z
Z
hV D (gi ◦ f ), Gi dkV k →
hFj , Gi dkV k as i → ∞
E(j)
E(j)
whenever G ∈ L∞ (kV k, Hom(Rn , Z)∗ ) and j ∈ P . Noting Ej ⊂ Ej+1 and
Fj (x) = Fj+1 (x) for kV k almost all E(j) for j ∈ P , one may define a kV k
measurable function F by F (x) = limj→∞ Fj (x) whenever x ∈ U .
Weakly Differentiable Functions on Varifolds
1027
To verify g ◦ f ∈ T(V , Z) with V D (g ◦ f )(x) = F (x) for kV k almost
all x , suppose θ ∈ D (U, Rn ) and γ ∈ E (Z, R) with spt D γ compact. Then,
there exists j ∈ P with spt θ ⊂ Kj and (g|B)−1 [spt D γ] ⊂ U(0, j). Define
Gi , G ∈ L∞ (kV k, Hom(Rn , Z)∗ ) by the requirements
hσ , Gi (x)i = hθ(x), D γ(gi (f (x))) ◦ σ i,
hσ , G(x)i = hθ(x), D γ(g(f (x))) ◦ σ i
whenever x ∈ dmn f and σ ∈ Hom(Rn , Z); hence,
kV k(∞) (Gi − G) → 0
as i → ∞.
Observing (gi |B)−1 [spt D γ] ⊂ U(0, j) for large i, one infers
Z
Z
hθ(x), D γ(g(f (x))) ◦ F (x)i dkV kx =
hF , Gi dkV k
E(j)
Z
= lim
hV D (gi ◦ f )(x), Gi (x)i dkV kx
i→∞ E(j)
Z
= lim hθ(x), D γ(gi (f (x)) ◦ V D (gi ◦ f )(x)i dkV kx
as hF , Gi(x) = 0 = hV D (gi ◦ f ), Gi i(x) for kV k almost all x ∈ U ∼ E(j).
and
❐
i→∞
Remark 8.16. Taking Υ = Y and g(y) = |y| for y ∈ Y yields |f | ∈ T(V )
kV D |f |(x)k ≤ kV D f (x)k
for kV k almost all x.
The following approximation of the identity map will prove useful.
Lemma 8.17. Whenever Y is a finite-dimensional normed vector space, there
exists a family of functions gs ∈ D (Y , Y ) with 0 < s < ∞ satisfying
gs (y) = y
whenever y ∈ Y ∩ B(0, s), 0 < s < ∞,
sup{Lip gs : 0 < s < ∞} < ∞.
Proof. One may assume Y = Rℓ for some ℓ ∈ P . Select ω ∈ D (R, R) such
that
0 ≤ ω(t) ≤ t
ω(t) = t
for 0 ≤ t < ∞,
for − 1 ≤ t ≤ 1,
define ωs = sω ◦ µ1/s and gs ∈ D (Y , Y ) by
gs (y) = 0
if y = 0,
gs (y) = ωs (|y|)|y|−1 y
if y ≠ 0,
U LRICH M ENNE
1028
whenever y ∈ Y and 0 < s < ∞, and conclude that
ωs (t) = t
for − s ≤ t ≤ s,
Lip ωs = Lip ω,
D gs (y)(v) = ω′s (|y|)(|y|−1 y) • v(|y|−1 y)
whenever y ∈ Y ∼{0}, v ∈ Y , and 0 < s < ∞. Hence, Lip gs ≤ 2 Lip ω < ∞.
❐
+ ωs (|y|)|y|−1 (v − (|y|−1 y) • v(|y|−1 y))
Lemma 8.18. Suppose that m, n, U , V , and Y are as in Definition 8.3,
f ∈ T(V , Y ) ∩ Lloc
1 (kV k + kδV k, Y ),
n
V D f ∈ Lloc
1 (kV k, Hom(R , Y )),
and α ∈ Hom(Y , R).
Then, α ◦ f ∈ T(V ) and
V D (α ◦ f )(x) = α ◦ V D f (x) for kV k almost all x,
Z
(P♮ • D θ(x))f (x) + hθ(x), V D f (x)i dV (x, P )
(δV )((α ◦ f )θ) = α
whenever θ ∈ D (U, Rn ).
❐
Proof. If f is bounded, the conclusion follows from Remark 8.6. The general
case may be treated by approximation based on Lemmas 8.12 and 8.17.
Remark 8.19. If ℓ = 1, m = n, and kV k(A) = L m (A) for A ⊂ U , then
1,1
f ∈ T(V ) if and only if f belongs to the class Tloc (U) introduced by Bénilan,
Boccardo, Gallouët, Gariepy, Pierre, and Vázquez in [BBG+ 95, p. 244]; in this
1,1
(U)” of [BBG+ 95,
case, V D f corresponds to “the derivative Df of f ∈ Tloc
p. 246] as may be verified by use of Lemma 8.12, Example 8.13 (1) (4), Lemma
8.18, and [BBG+ 95, 2.1, 2.3].
The next theorem concerns a measure-theoretic locality property of the generalised weak derivative, and the join, addition, and multiplication of a generalised
weakly differentiable function and a locally Lipschitzian function.
Theorem 8.20. Suppose m, n, U , V , and Y are as in Definition 8.3, and
f ∈ T(V , Y ).
Then, the following four statements hold:
(1) If A = {x : f (x) = 0}, then V D f (x) = 0 for kV k almost all x ∈ A.
(2) If Z is a finite-dimensional normed vector space, g : U → Z is locally Lipschitzian, and h(x) = (f (x), g(x)) for x ∈ dmn f , then h ∈ T(V , Y ×Z)
and
V D h(x)(u) = (V D f (x)(u), V D g(x)(u))
for kV k almost all x .
whenever u ∈ Rn
Weakly Differentiable Functions on Varifolds
1029
(3) If g : U → Y is locally Lipschitzian, then f + g ∈ T(V , Y ) and
V D (f + g)(x) = V D f (x) + V D g(x)
for kV k almost all x.
loc
n
(4) If f ∈ Lloc
1 (kV k, Y ), V D f ∈ L1 (kV k, Hom(R , Y )), and g : U → R is
locally Lipschitzian, then gf ∈ T(V , Y ) and
V D (gf )(x) = V D g(x)f (x) + g(x)V D f (x)
for kV k almost all x.
Proof of (1). By Lemma 8.17 in conjunction with Lemma 8.12, one may asn
sume f to be bounded and V D f ∈ Lloc
1 (kV k, Hom(R , Y )); hence, by Lemma
8.18, also Y = R. In this case, it follows from Lemma 8.12 and Example 8.13 (1) (4)
that f + and f − satisfy the same hypotheses as f ; hence,
(δV )(gθ) =
Z
(P♮ • D θ(x))g(x) + hθ(x), V D g(x)i dV (x, P )
for θ ∈ D (U, R) and g ∈ {f , f +, f − } by Lemma 8.18. Since f = f + − f − , this
implies
for kV k almost all x,
and the formulae derived in Example 8.13 (4) yield the conclusion.
❐
V D f (x) = V D f + (x) − V D f − (x)
Proof of (2). Assume dim Y > 0 and dim Z > 0; hence, dim(Y × Z) ≥ 2.
Define a kV k measurable function H with values in Hom(Rn , Y × Z) by
H(x)(u) = (V D f (x)(u), V D g(x)(u))
for u ∈ Rn
whenever x ∈ dmn V D f ∩ dmn V D g . It will be proven that
(δV )((γ ◦ h)θ) =
Z
γ(h(x))P♮ • D θ(x) + hθ(x), D γ(h(x)) ◦ H(x)i dV (x, P )
whenever γ ∈ D (Y × Z, R) and θ ∈ D (U, Rn ); in fact, in view of Remark 2.15
and Lemma 3.1, the problem reduces to the case that, for some µ ∈ D (Y , R) and
some ν ∈ D (Z, R), we have γ(y, z) = µ(y)ν(z) for (y, z) ∈ Y × Z, in which
case, one computes, using Remark 8.6 and Example 8.7, that
=
Z
(δV )((γ ◦ h)θ) = (δV )((µ ◦ f )(ν ◦ g)θ)
µ(f (x)) hθ(x), D ν(g(x)) ◦ V D g(x)i + ν(g(x))P♮ • D θ(x) dV (x, P )
Z
+ ν(g(x))hθ(x), D µ(f (x)) ◦ V D f (x)i dkV kx
γ(h(x))P♮ • D θ(x) + hθ(x), D γ(h(x)) ◦ H(x)i dV (x, P ).
The conclusion now follows from Remark 8.4.
❐
=
Z
U LRICH M ENNE
1030
Proof of (3). Assume dim Y > 0 and that im g ⊂ B(0, t) for some 0 < t < ∞.
Define h as in (2), and let L : Y × Y → Y denote addition. The following assertion
will be shown: if γ ∈ D (Y , R) and θ ∈ D (U, Rn ), then
(δV )((γ ◦ L ◦ h)θ)
Z
= γ(L(h(x)))P♮ • D θ(x) + hθ(x), D(γ ◦ L)(h(x)) ◦ V D h(x)i dV (x, P ).
For this purpose, define D = Y × (Y ∩ B(0, 2t)), and choose ̺ ∈ D (Y , R) with
B(0, t) ⊂ Int{z : ̺(z) = 1},
spt ̺ ⊂ B(0, 2t).
Let ϕ : Y × Y → Y be defined by
ϕ(y, z) = ̺(z)(γ ◦ L)(y, z)
for (y, z) ∈ Y × Y .
Noting that
spt ϕ ⊂ D ∩ L−1 [spt γ], spt D ϕ is compact,
ϕ(y, z) = (γ ◦ L)(y, z) and D ϕ(y, z) = D(γ ◦ L)(y, z)
L|D is proper,
for y ∈ Y and z ∈ Y ∩ B(0, t), one uses (2) and Definition 8.3 (2) with f and γ
replaced by h and ϕ to infer the assertion.
If dim Y ≥ 2 or f is bounded, the conclusion now follows from the assertion
of the preceding paragraph in conjunction with Remark 8.4. Finally, to approximate f in case dim Y = 1, one assumes Y = R and employs the functions fi
defined by fi (x) = sup{inf{f (x), i}, −i} for x ∈ dmn f and i ∈ P , and notices
that fi ∈ T(V ) and
|V D fi (x)| ≤ |V D f (x)|,
V D fi (x) → V D f (x)
implies V D fi (x) = 0
for kV k almost all x by Lemma 8.12, and Example 8.13 (1) (4).
❐
|fi (x)| < |f (x)|
as i → ∞
as i → ∞,
❐
Proof of (4). Assume g is bounded, define h as in (2), and let µ : Y × R → Y
be defined by µ(y, t) = ty for y ∈ Y and t ∈ R. If f is bounded, the conclusion
follows from (2) and Remark 8.6 with f and g replaced by h and µ. The general
case then follows by approximation using Lemma 8.12 and Lemma 8.17.
Remark 8.21. The approximation procedure in the proof of (2) uses ideas
from [Fed69, 4.1.2, 4.1.3].
Remark 8.22. The need for some strong restriction on g in (2)–(4) will be
illustrated in Example 8.25.
Weakly Differentiable Functions on Varifolds
1031
Note 8.23. If ϕ is a measure, A is ϕ measurable, and B is a ϕ x A measurable
subset of A, then B is ϕ measurable.
Generalised weakly differentiable functions integrate well with the concept
of decomposition of a varifold. Under a natural summability hypothesis on its
derivative, a generalised weakly differentiable function may be defined on each
component separately.
Theorem 8.24. Suppose m, n, U , V , and Y are as in Definition 8.3, Ξ is a
decomposition of V , ξ is associated with Ξ as in Remark 6.10, fW ∈ T(W , Y ) for
W ∈ Ξ, and
f =
F=
[
[
{fW |ξ(W ) : W ∈ Ξ},
{W D fW |ξ(W ) : W ∈ Ξ}.
Then, the following three statements hold:
(1) f is kV k + kδV k measurable.
(2) F is
Z kV k measurable.
(3) If
kF k dkV k < ∞ whenever K is a compact subset of U and
K∩{x : |f (x)|≤s}
0 ≤ s < ∞, then f ∈ T(V , Y ) and V D f (x) = F (x) for kV k almost all x .
Proof. Clearly, we have f |ξ(W ) = fW |ξ(W ) and F |ξ(W ) = W D fW |ξ(W )
for W ∈ Ξ. Then, (1) and (2) follow by Note 8.23, since kW k = kV k x ξ(W ) and
kδW k = kδV k x ξ(W ) for W ∈ Ξ. The hypothesis of (3) implies
(δV )((γ ◦ f )θ) =
=
X Z
X
(δW )((γ ◦ fW )θ)
W ∈Ξ
γ(fW (x))P♮ • D θ(x)
W ∈Ξ
+ hθ(x), D γ(fW (x)) ◦ W D fW (x)i dW (x, P )
W ∈Ξ ξ(W )×G(n,m)
=
Z
γ(f (x))P♮ • D θ(x)
+ hθ(x), D γ(f (x)) ◦ F (x)i dV (x, P )
γ(f (x))P♮ • D θ(x) + hθ(x), D γ(f (x)) ◦ F (x)i dV (x, P )
whenever θ ∈ D (U, Rn ), γ ∈ E (Y , R) and spt D γ is compact.
❐
=
X Z
Based on the nonuniqueness of decompositions, we obtain an example showing that the join, the sum, or the multiplication of two generalised weakly differentiable functions need not be generalised weakly differentiable even if their
derivatives vanish.
U LRICH M ENNE
1032
Example 8.25. Suppose Rj , Vj , and V are as in Remark 6.13, and define
f : C → R, g : C → R, and h : C → R by
f (x) =
(
1 if x ∈ R1 ∪ R3 ∪ R5 ,
0 otherwise,
g(x) =
(
1 if x ∈ R1 ∪ R4 ,
0 otherwise,
h(x) = (f (x), g(x)),
whenever x ∈ C.
Then, f ∈ T(V1 + V3 + V5 ), g ∈ T(V1 + V4 ); hence, f , g ∈ T(V ) with
V D f (x) = 0 = V D g(x)
for kV k almost all x
by Theorem 8.24. However, neither h ∉ T(V , R × R) nor f + g ∈ T(V ) nor
gf ∈ T(V ); in fact, the characteristic function of R1 that equals gf does not
belong to T(V ); hence, f + g ∉ T(V ) by Lemma 8.12 and h ∈ T(V , R × R)
would imply f + g ∈ T(V ) by Remark 8.6 with f and g replaced by h and the
addition on R.
Remark 8.26. Modifying V outside a neighbourhood of 0, one may construct
similar examples for indecomposable varifolds.
Remark 8.27. Here, we discuss some properties of the class W(V , Y ). Whenever m, n, U , V , and Y are as in Definition 8.3, the class is defined as consisting
loc
n
of all f ∈ Lloc
1 (kV k + kδV k, Y ) such that, for some F ∈ L1 (kV k, Hom(R , Y )),
(δV )((α ◦ f )θ) = α
Z
(P♮ • D θ(x))f (x) + hθ(x), F (x)i dV (x, P )
whenever θ ∈ D (U, Rn ) and α ∈ Hom(Y , R).
Clearly, F is kV k almost unique and W(V , Y ) is a vector space. Note that
loc
n
T(V , Y ) ∩ Lloc
1 (kV k + kδV k, Y ) ∩ {f : V D f ∈ L1 (kV k, Hom(R , Y ))}
is contained in W(V , Y ) by Lemma 8.18. However, it may happen that f ∈
W(V , Y ) but ̺ ◦ f ∉ W(V , Y ) for some ̺ ∈ D (Y , R); in fact, the function f + g
constructed in Example 8.25 provides an instance.
Example 8.28. The considerations of Example 8.25 may be axiomatised as
follows.
Suppose Φ denotes the family of stationary one-dimensional integral varifolds
in R2 and whenever V ∈ Φ, C(V ) is a class of real-valued functions on R2 satisfying the following conditions:
Weakly Differentiable Functions on Varifolds
1033
(1) If Ξ is a decomposition of V , ξ is associated wtih Ξ as in Remark 6.10,
υ : Ξ → R, and f : R2 → R satisfies
f (x) =
(
υ(W )
0
if x ∈ ξ(W ), W ∈ Ξ,
S
if x ∈ R2 ∼ im ξ,
then f ∈ C(V ).
(2) If f , g ∈ C(V ), then f + g ∈ C(V ).
(3) If f ∈ C(V ), ω ∈ E (R, R) and spt ω′ is compact, then ω ◦ f ∈ C(V ).
Then, using the terminology of Remark 6.13, the characteristic function of any
ray Rj belongs to C(V ). Moreover, the same holds if the conditions (2) and (3)
are replaced by the following condition:
(4) If f , g ∈ C(V ), then f g ∈ C(V ).
To prepare for the coarea formula of this section, a formula of the distributional boundary of superlevel sets of a generalised weakly differentiable function
will be given.
Lemma 8.29. Suppose m, n, U , and V are as in Definition 8.3, f ∈ T(V ),
y ∈ R, and E = {x : f (x) > y}.
Z
hθ, V D f i dkV k for
Then, we have that V ∂E(θ) = lim ε−1
ε→0+
{x : y<f (x)≤y+ε}
θ ∈ D (U, Rn ).
Proof. Suppose θ ∈ D (U, Rn ). Define gε : R → R by
gε (υ) = ε−1 inf{ε, sup{υ − y, 0}}
whenever υ ∈ R and 0 < ε ≤ 1. Notice that
gε (υ) = 0
gε (υ) = 1
gε (υ) ↑ 1
if υ ≤ y,
if υ ≥ y + ε,
as ε → 0 + if υ > y
whenever υ ∈ R. Consequently, one infers gε ◦ f ∈ T(V ) with
V D (gε ◦ f )(x) =
(
ε−1 V D f (x)
V D (gε ◦ f )(x) = 0
if y < f (x) ≤ y + ε,
otherwise
for kV k almost all x from Lemma 8.12, Example 8.13 (1) (2) (4), and Theorem 8.20 (1). It follows that
Z
V ∂E(θ) = lim (δV )((gε ◦ f )θ) − lim (gε ◦ f )(x)P♮ • D θ(x) dV (x, P )
ε→0+
ε→0+
Z
−1
hθ(x), V D f (x)i dkV kx,
= lim ε
{x : y<f (x)≤y+ε}
where Lemma 8.18 with α and f replaced by 1R and gε ◦ f was employed.
❐
ε→0+
U LRICH M ENNE
1034
Now, the first coarea formula for real-valued generalised weakly differentiable
function follows with ease.
Theorem 8.30. Suppose m, n, U , and V are as in Definition 8.3, f ∈ T(V ),
T ∈ D ′ (U × R, Rn ) satisfies
T (ϕ) =
Z
hϕ(x, f (x)), V D f (x)i dkV kx
for ϕ ∈ D (U × R, Rn ),
and E(y) = {x : f (x) > y} for y ∈ R.
Then, T is representable by integration and
Z
V ∂E(y)(ϕ(·, y)) dL 1 y,
Z
ZZ
g dkT k =
g(x, y) dkV ∂E(y)kx dL 1 y
T (ϕ) =
whenever ϕ ∈ L1 (kT k, Rn ) and g is an R̄-valued kT k integrable function. In particular, for L 1 almost all y , the distribution V ∂E(y) is representable by integration
and kV ∂E(y)k(U ∩ {x : f (x) ≠ y}) = 0.
Proof. Taking Remark 8.5 into account, Lemma 8.1 yields
Z
ω ◦ f hθ, V D f i dkV k = T(x,y) (ω(y)θ(x)) =
Z
ω(y)V ∂E(y)(θ) dL 1 y
❐
for ω ∈ D (R, R) and θ ∈ D (U, Rn ). In view of Lemma 8.29, the principal
conclusion is implied by Theorem 3.4 (1) (2) with J = R and Z = Rn . Therefore, the postscript follows by employing Remark 8.5 and also by noting that
(U × R) ∩ {(x, y) : f (x) ≠ y} is kT k measurable since f is kV k almost equal to
a Borel function by [Fed69, 2.3.6].
Remark 8.31. The formulation of Theorem 8.30 is modelled on [Fed69,
4.5.9 (13)].
Remark 8.32. The equalities in Theorem 8.30 are not valid for functions in
W(V ) with V D f in the definition of T replaced by the function F occurring in
the definition of W(V ) (see Remark 8.27); in fact, the function f + g constructed
in Example 8.25 provides an instance.
Corollary 8.33. Suppose m, n, U , V , and Y are as in Definition 8.3 and
f ∈ T(V , Y ).
Then, (kδV k x X)(∞) (f ) ≤ (kV k x X)(∞) (f ) whenever X is an open subset of U .
❐
Proof. Assume X = U and abbreviate s = kV k(∞) (f ). Recalling Remark 8.16,
one applies Theorem 8.30 with f replaced by |f | to infer V ∂E(t) = 0 for
L 1 almost all s < t < ∞, where E(t) = {x : |f (x)| > t}; hence, we have
kδV k(E(t)) = 0 for those t .
Weakly Differentiable Functions on Varifolds
1035
To conclude this section, it is shown that generalised weakly differentiable
functions are almost constant on the components of some decomposition; this in
particular restates and employs the existence of a decomposition already obtained
in Theorem 6.12. Additionally, the coarea formula (see Theorem 8.30) will be
useful in the proof.
Theorem 8.34. Suppose that m, n, U , V , and Y are as in Definition 8.3,
f ∈ T(V , Y ), and V D f (x) = 0 for kV k almost all x . Then, there exist a decomposition Ξ of V and a map υ : Ξ → Y such that f (x) = υ(W ) for kW k + kδW k almost
all x whenever W ∈ Ξ.
Proof. Define B(y, r ) = {x : |f (x) − y| ≤ r } for y ∈ Y and 0 ≤ r < ∞.
First, one observes that Theorem 8.30 in conjunction with Lemma 8.12, Example 8.13 (2), and Remark 8.16 implies V ∂B(y, r ) = 0 for y ∈ Y and 0 ≤ r < ∞.
Next, a countable subset Υ of Y such that f (x) ∈ Υ for kV k almost all x will
be constructed. For this purpose, define δi = α(m)2−m−1 i−1−2m , εi = 2−1 i−2 ,
and Borel sets Ai consisting of all a ∈ Rn satisfying
|a| ≤ i,
U(a, 2εi ) ⊂ U,
kδV k B(a, s) ≤ α(m)is m
Θ m (kV k, a) ≥
1
i
,
for 0 < s < εi
S
whenever i ∈ P . Clearly, Ai ⊂ Ai+1 for i ∈ P and kV k(U ∼ ∞
i=1 Ai ) = 0 by Allard [All72, 3.5 (1a)] and [Fed69, 2.8.18, 2.9.5]. Abbreviating µi = f# (kV k x Ai ),
one obtains
kV k(B(y, r ) ∩ {x : dist(x, Ai ) ≤ εi }) ≥ δi
whenever y ∈ spt µi , 0 ≤ r < ∞, and i ∈ P ; in fact, if we assume r > 0,
there exists a ∈ Ai with Θ m (kV k x B(y, r ), a) = Θ m (kV k, a) ≥ 1/i by [Fed69,
2.8.18, 2.9.11], implying
Z εi
0
s −m kδ(V x B(y, r ) × G(n, m))k B(a, s) dL 1 s ≤ α(m)iεi ,
whence we infer kV k(B(y, r ) ∩ B(a, εi )) ≥ δi by Corollary 4.5 and Remark 4.6.
As {B(y, 0) : y ∈ spt µi } is disjointed, one deduces
δi card spt µi ≤ kV k(U ∩ {x : dist(x, Ai ) ≤ εi }) < ∞,
S
and one may take Υ = ∞
i=1 spt µi , since µi (Y ∼ spt µi ) = 0.
Applying Theorem 6.12 to V x B(y, 0) × G(n, m) for y ∈ Υ , and recalling
Remarks 5.4 and 6.10, one constructs a decomposition Ξ of V and a function ξ
mapping Ξ into the class of all kV k + kδV k measurable sets such that distinct
members of Ξ are mapped onto disjoint sets, such that
W = V x ξ(W ) × G(n, m),
V ∂ξ(W ) = 0
U LRICH M ENNE
1036
(kW k + kδW k)(U ∼ ξ(W )) = 0
the conclusion is now evident.
for W ∈ Ξ,
❐
whenever W ∈ Ξ, and such that each ξ(W ) is contained in some B(y, 0). If we
define υ : Ξ → Y by the requirement {υ(W )} = f [ξ(W )] for W ∈ Ξ, and note
Remark 8.35. Clearly, the second paragraph of the proof has conceptual overlap with the second and third paragraph of the proof of Theorem 6.12.
9. Z ERO B OUNDARY VALUES
Introduction. In this section, a notion of zero boundary values for nonnegative generalised weakly differentiable functions based on the behaviour of superlevel sets is introduced. Here, stability of this class under composition (see Lemma
9.9), convergence (see Corollary 9.13 and Remark 9.14), and multiplication by
a nonnegative Lipschitzian function (see Theorem 9.16) are investigated. The
deeper parts of this study rest on a characterisation of such functions in terms of
an associated distribution built from certain distributional boundaries of superlevel sets (see Theorem 9.12).
First, the appropriate class of functions will be defined.
Definition 9.1. Suppose that m, n ∈ P , m ≤ n, U is an open subset of Rn ,
V ∈ RV m (U), kδV k is a Radon measure, and G is a relatively open subset of
Bdry U . Then, TG (V ) is defined to be the set of all nonnegative functions f ∈
T(V ) such that with
B = (Bdry U) ∼ G and E(y) = {x : f (x) > y}
for 0 < y < ∞
the following conditions hold for L 1 almost all 0 < y < ∞ (see Definitions 2.18
and 2.20 and Remark 2.19):
Z
(kV k + kδV k)(E(y) ∩ K) + kV ∂E(y)k(U ∩ K) < ∞,
E(y)×G(n,m)
P♮ • D θ(x) dV (x, P ) = ((δV ) x E(y))(θ|U) − V ∂E(y)(θ|U)
whenever K is a compact subset of Rn ∼ B and θ ∈ D (Rn ∼ B, Rn ).
Remark 9.2. Notice that |f | ∈ T0 (V ) whenever Y is a finite-dimensional
normed vector space and f ∈ T(V , Y ) by Remark 8.16 and Theorem 8.30.
Remark 9.3. The condition on E(y) has been studied in Section 7 under
the supplementary hypothesis on V that Θ m (kV k, x) ≥ 1 for kV k almost all x
(see Hypothesis 7.1). In this section, this hypothesis will not occur, leaving those
properties of TG (V ) employing the additional hypothesis on V to Section 10.
The next two auxiliary facts confirm two basic properties for use in the Sobolev
Poincaré type estimates in Theorem 10.1. Recall that the support of a measure µ
over an open subset U of Rn is by definition a relatively closed subset of U ; its
closure in Rn may be larger.
Weakly Differentiable Functions on Varifolds
1037
Lemma 9.4. Suppose m, n, U , V , G, B , and E(y) are as in Definition 9.1,
f ∈ TG (V ), and
Clos spt((kV k + kδV k) x E(y)) ⊂ Rn ∼ B
for L 1 almost all 0 < y < ∞.
Then, f ∈ TBdry U (V ).
Proof. Define A(y) = Clos spt((kV k + kδV k) x E(y)) for 0 < y < ∞. Suppose y satisfies the conditions of Definition 9.1 and A(y) ⊂ Rn ∼ B . One concludes
(kV k + kδV k)(E(y) ∩ K) + kV ∂E(y)k(U ∩ K) < ∞
whenever K is a compact subset of Rn . Hence, one may define S ∈ D ′ (Rn , Rn )
by
S(θ) =
Z
E(y)×G(n,m)
P♮ • D θ(x) dV (x, P )
− ((δV ) x E(y))(θ|U) + V ∂E(y)(θ|U)
❐
whenever θ ∈ D (Rn , Rn ). Notice that spt S ⊂ A(y) ⊂ Rn ∼ B . On the other
hand the conditions of Definition 9.1 imply spt S ⊂ B . It follows that spt S = 0
and S = 0.
Lemma 9.5. Suppose that m, n, U , V , G, and B are as in Definition 9.1,
f ∈ TG (V ), X is an open subset of Rn ∼ B , H = X ∩ Bdry U , and
W = V |2(U ∩X)×G(n,m) .
Then, f |X ∈ TH (W ).
Proof. Notice that H = X ∩ Bdry(U ∩ X). In particular, H is a relatively open
subset of Bdry(U ∩ X). Let C = (Bdry(U ∩ X)) ∼ H , and observe the inclusions
(Rn ∼ C) ∩ Clos(U ∩ X) ⊂ X ⊂ Rn ∼(B ∪ C).
Suppose that 0 < y < ∞ satisfies the conditions of Definition 9.1. Define
E = {x : f (x) > y}, and notice that
(kW k + kδW k)(E ∩ X ∩ K) + kW ∂(E ∩ X)k(U ∩ X ∩ K) < ∞
whenever K is a compact subset of Rn ∼ C by the first inclusion of the first paragraph. Therefore, one may define S ∈ D ′ (Rn ∼ C, Rn ) by
S(θ) =
Z
(E∩X)×G(n,m)
P♮ • D θ(x) dW (x, P )
− ((δW ) x E ∩ X)(θ|U ∩ X) + (W ∂(E ∩ X))(θ|U ∩ X)
U LRICH M ENNE
1038
whenever θ ∈ D (Rn ∼ C, Rn ). Notice that spt S ⊂ (Rn ∼ C) ∩ Clos(U ∩ X) ⊂ X .
On the other hand, the conditions of Definition 9.1 in conjunction with the
second inclusion of the first paragraph imply
S(θ|Rn ∼ C) =
Z
E×G(n,m)
P♮ • D θ(x) dV (x, P )
− ((δV ) x E)(θ|U) + V ∂E(θ|U) = 0
❐
whenever θ ∈ D (Rn , Rn ) and spt θ ⊂ X ; hence, X ∩ spt S = 0. It follows that
spt S = 0 and S = 0.
Remark 9.6. Recalling the first paragraph of the proof of Lemma 9.5, one
notices that
U ∩ {x : U(x, r ) ⊂ X} ⊂ U ∩ X ∩ {x : U(x, r ) ⊂ Rn ∼ C}
for 0 < r < ∞;
this fact will be useful for localisation in Theorem 10.1 (1d).
Example 9.7. Suppose m = n = 1, U = R ∼{0}, V ∈ RV 1 (U) with
kV k(A) = L 1 (A) for A ⊂ U , f : U → R with f (x) = 1 for x ∈ U , X =
U ∩ {x : x < 0}, and W = V |2X×G(1,1) .
Then, δV = 0 and f ∈ T{0} (V ), but f |X ∉ T{0} (W ).
Remark 9.8. The preceding example shows that Lemma 9.5 may not be
sharpened by allowing H to be an arbitrary relatively open subset of Bdry(U ∩ X)
contained in G.
We note that the favourable stability properties of elements of T(V ) under
composition by suitable Lipschitzian functions are shared by members of TG (V ).
Lemma 9.9. Suppose m, n, U , V , and G are as in Definition 9.1, f ∈ TG (V ),
Υ is a closed subset of {y : 0 ≤ y < ∞}, and g : {y : 0 ≤ y < ∞} → {z : 0 ≤ z < ∞}
is a Lipschitzian function such that g(0) = 0, g|Υ is proper, and g|R ∼ Υ is locally
constant. Then, g ◦ f ∈ TG (V ).
Proof. Let h = g ∪ {(y, 0) : −∞ < y < 0}. Since Lip h = Lip g and h|R ∼ Υ
is locally constant, Lemma 8.15 yields g ◦ f = h ◦ f ∈ T(V ).
Let B = (Bdry U) ∼ G. Define F to be the class of all Borel subsets Y of
{y : 0 < y < ∞} such that E = f −1 [Y ] satisfies
Z
E×G(n,m)
(kV k + kδV k)(E ∩ K) + kV ∂Ek(U ∩ K) < ∞,
P♮ • D θ(x) dV (x, P ) = ((δV ) x E)(θ|U) − (V ∂E)(θ|U)
whenever K is a compact subset of Rn ∼ B S
and θ ∈ D (Rn ∼ B,SRn ). If Y ∈ F ,
H is a finite disjointed subfamily of F and H ⊂ Y , then Y ∼ H ∈ F , as one
readily verifies using Remark 5.3. Let O denote the set of 0 < b < ∞ such
that either {y : b < y < ∞} or {y : b ≤ y < ∞} does not belong to F . Since
Weakly Differentiable Functions on Varifolds
1039
(kV k + kδV k)(f −1 [{b}]) = 0 for all but countably many b , one obtains that
L 1 (O) = 0.
Next, it will be shown that D = g −1 [{z : c < z < ∞}] ∈ F whenever c
satisfies 0 < c ∈ R ∼ g[O] and N(g, c) < ∞. For this purpose, assume D ≠ 0.
Since inf D > 0 and Bdry D ⊂ {y : g(y) = c}, one infers that Bdry D is a finite
subset of {b : 0 < b < ∞} ∼ O ; in particular, Y = {y : inf D < y < ∞} ∈ F .
Let H denote the family of connected components of Y ∼ D . Observe that H is a
finite disjointed family of possibly degenerated closed intervals. Since
Bdry I ⊂ Bdry D,
I = {y : inf I ≤ y < ∞} ∼{y : sup I < y < ∞}
S
whenever I ∈ H , it follows that H ⊂ F and D = Y ∼ H ∈ F .
Finally, notice that L 1 almost all 0 < c < ∞ satisfy
for i ∈ P by [Fed69, 3.2.3 (1)], and so ∞ > N(g|Υ, c) = N(g, c) for such c .
❐
N(g|{y : y ≤ i}, c) < ∞
Recall that sign : R → R satisfies sign(x) = |x|−1 x for 0 ≠ x ∈ R and
sign(0) = 0.
Example 9.10. Suppose m = n = 1, U = R ∼{0}, and V ∈ RV m (U) with
kV k(A) = L 1 (A) for A ⊂ U .
Then, δV = 0, and the following two statements hold:
(1) If f = sign |U , then f ∈ T(V ) and |f | ∈ T{0} (V ), but f + ∉ T{0} (V ).
(2) If y, b ∈ R2 , |y| = |b|, ν is a norm on R2 , ν(y) ≠ ν(b), f (x) = y
for 0 > x ∈ R and f (x) = b for 0 < x ∈ R, then f ∈ T(V , R2 ) and
|f | ∈ T{0} (V ), but ν ◦ f ∉ T{0} (V ).
Remark 9.11. The preceding example shows that the class
T(V , Y ) ∩ {f : |f | ∈ TG (V )}
where Y is a finite-dimensional normed vector space, does not satisfy stability
properties similar to those proven for TG (V ) in Lemma 9.9.
With the functional analytic machinery from Sections 2 and 3, and also the
measure-theoretic coarea formula (see Remark 8.5 and Theorem 8.30), a useful
characterisation of elements of TG (V ) satisfying a mild summability hypothesis
on their derivative near G may be obtained.
Theorem 9.12. Suppose that m, n, U , V , G, and B are as in Definition 9.1,
f ∈ T(V ), f is nonnegative, J = {y : 0 < y < ∞}, E(y) = U ∩ {x : f (x) > y}
for y ∈ J ,
(kV k + kδV k)(K ∩ E(y)) < ∞
U LRICH M ENNE
1040
whenever K is a compact subset of Rn ∼ B and y ∈ J , and the distributions
S(y) ∈ D ′ (Rn ∼ B, Rn ) and T ∈ D ′ ((Rn ∼ B) × J, Rn ) satisfy
S(y)(θ) = ((δV ) x E(y))(θ|U)
Z
−
P♮ • D θ(x) dV (x, P )
E(y)×G(n,m)
Z
T (ϕ) = S(y)(ϕ(·, y)) dL 1 y
for y ∈ J,
J
whenever θ ∈ D (Rn ∼ B, Rn ) and ϕ ∈ D ((Rn ∼ B) × J, Rn ).
Then, the following three conditions are equivalent:
(1)
Z
K∩{x : f (x)∈I}
n
|V D f (x)| dkV kx < ∞ whenever K is a compact subset of
R
Z ∼ B and I is a compact subset of J , and f ∈ TG (V ).
(2) kS(y)k(K) dL 1 y < ∞ whenever K is a compact subset of Rn ∼ B and I
I
is a compact subset of J , and kS(y)k((Rn ∼ B) ∼ U) = 0 for L 1 almost all
y ∈ J.
(3) T is representable by integration and kT k(((Rn ∼ B) ∼ U) × J) = 0.
If these conditions are satisfied, then the following statements with A = f −1[J] hold:
(4) For L 1 almost all y ∈ J , there holds
whenever θ ∈ D (Rn ∼ B, Rn ).
S(y)(θ) = V ∂E(y)(θ|U)
(5) If ϕ ∈ L1 (kT k, Rn), then
T (ϕ) =
Z
1
J
V ∂E(y)(ϕ(·, y)|U) dL y =
Z
A
hϕ(x, f (x)), V D f (x)i dkV kx.
(6) If g is an R̄-valued kT k integrable function, then
Z
g dkT k =
=
Z Z
Z
J
A
g(x, y) dkV ∂E(y)kx dL 1 y
g(x, f (x))|V D f (x)| dkV kx.
Proof. Notice that S(y)(θ) = V ∂E(y)(θ|U) whenever θ ∈ D (Rn ∼ B, Rn )
and spt θ ⊂ U . From Remark 8.5 and Theorem 8.30, one infers
Z
K∩{x : f (x)∈I}
|V D f (x)| dkV kx =
Z
I
kV ∂E(y)k(U ∩ K) dL 1 y
whenever K is a compact subset of Rn ∼ B and I is a compact subset of J . Consequently, (1) and (2) are equivalent.
Weakly Differentiable Functions on Varifolds
1041
Moreover, one remarks that S is L 1 x J measurable by Example 2.24, and,
employing a countable sequentially dense subset of D (Rn ∼ B, Rn ), one obtains
that, for L 1 almost all y ∈ J ,
S(y)(θ) = lim ε−1
ε→0+
Z y+ε
whenever θ ∈ D (Rn ∼ B, Rn )
S(υ)(θ) dL 1 υ
y
by Remarks 2.2, 2.15, and [Fed69, 2.8.17, 2.9.8]. Defining Rθ ∈ D ′ (J, R) by
whenever ω ∈ D (J, R) and θ ∈ D (Rn ∼ B, Rn ),
Rθ (ω) = T(x,y) (ω(y)θ(x))
one then notes that kRθ k is absolutely continuous with respect to L 1 |2J for
θ ∈ D (Rn ∼ B, Rn ). If (2) holds, then T is representable by integration, and
Theorem 3.4 (2) with U and Z replaced by Rn ∼ B and Rn yields (3). Conversely,
if (3) holds, then (2) follows similarly from Theorem 3.4 (1).
Suppose now (1)–(3) hold. Then, (4) is evident from Definition 9.1, and
implies
T (ϕ) =
Z
A
hϕ(x, f (x)), V D f (x)i dkV kx
for ϕ ∈ D ((Rn ∼ B) × J, Rn)
❐
by Remark 8.5 and Theorem 8.30. Finally, (5) and (6) follow from Lemma 3.3
and Theorem 3.4 (2) with U replaced by Rn ∼ B .
As a first application of the preceding characterisation, the following closedness result for TG (V ) in T(V ) is obtained.
Corollary 9.13. Suppose that m, n, U , V , G, and B are as in Definition 9.1,
that f ∈ T(V ) is a nonnegative function, that fi is a sequence in TG (V ), that
J = {y : 0 < y < ∞}, and that
(kV k + kδV k)(K ∩ {x : f (x) > b}) < ∞,
fi → f
as i → ∞ in (kV k + kδV k) x U ∩ K measure,
̺(K, I, b, δ) < ∞ for 0 < δ < ∞,
̺(K, I, b, δ) → 0 as δ → 0+
whenever K is a compact subset of Rn ∼ B , I is a compact subset of J , and inf I > b ∈
J , where ̺(K, I, b, δ) denotes the supremum of all numbers
lim sup
i→∞
Z
K∩A∩{x : fi (x)∈I}
|V D fi | dkV k
corresponding to kV k measurable sets A with kV k(A ∩ K ∩ {x : f (x) > b}) ≤ δ.
Then, f ∈ TG (V ).
U LRICH M ENNE
1042
Proof. Define
C = (U × J) ∩ {(x, y) : f (x) > y},
Ci = (U × J) ∩ {(x, y) : fi (x) > y}
for i ∈ P , E(y) = {x :(x, y) ∈ C} and Ei (y) = {x :(x, y) ∈ Ci } for y ∈ J
and i ∈ P , and g : (U × G(n, m)) × J → U × J by g((x, P ), y) = (x, y) for
x ∈ U , P ∈ G(n, m), and y ∈ J . As in Lemma 8.1, one derives that C and Ci
are kδV k × L 1 measurable and that g −1 [C] and g −1 [Ci ] are V × L 1 measurable.
Defining T ∈ D ′ (Rn ∼ B, Rn ) as in Theorem 9.12, Fubini’s theorem yields that
(see Hypothesis 4.4)
T (ϕ) =
Z
C
η(V , x) • ϕ(x, y) d(kδV k × L 1 )(x, y)
Z
−
P♮ • D ϕ(·, y)(x) d(V × L 1 )((x, P ), y)
g −1 [C]
for ϕ ∈ D ((Rn ∼ B) × J, Rn ). Observe that
(kV k + kδV k) × L 1 ((U ∩ K) × I) ∩ C < ∞,
lim (kV k + kδV k) × L 1 ((U ∩ K) × I) ∩ ((C ∼ Ci ) ∪ (Ci ∼ C)) = 0
i→∞
whenever K is a compact subset of U and I a compact subset of J ; in fact, if
b, ε ∈ J , b + ε < inf I , and A = {x : |f (x) − fi (x)| > ε}, then
(U × I) ∩ (C ∼ Ci ) ⊂ (A × I) ∪ {(x, y) : b < f (x) − ε ≤ y < f (x)},
(U × I) ∩ (Ci ∼ C) ⊂ (A × I) ∪ {(x, y) : b < f (x) ≤ y < f (x) + ε}.
Consequently, one employs Fubini’s theorem and Definition 9.1 to compute
T (ϕ) = lim
i→∞
Z
Ci
η(V , x) • ϕ(x, y) d(kδV k × L 1 )(x, y)
Z
−
P♮ • D ϕ(·, y)(x) d(V × L 1 )((x, P ), y)
g −1 [C ]
Z i
((δV ) x Ei (y))(ϕ(·, y)|U)
= lim
i→∞ J
Z
−
P♮ • D ϕ(·, y)(x) dV (x, P ) dL 1 y
Ei (y)×G(n,m)
Z
= lim V ∂Ei (y)(ϕ(·, y)|U) dL 1 y.
i→∞ J
In view of Theorem 8.30, one infers kT k(((Int K) ∼ A) × Int I) ≤ ̺(K, I, b, δ)
whenever K is a compact subset of Rn ∼ B , I is a compact subset of J , inf I > b ∈
J , 0 < δ < ∞, A is a compact subset of U , and
kV k((K ∼ A) ∩ E(b)) ≤ δ.
Weakly Differentiable Functions on Varifolds
1043
❐
In particular, taking A = 0 and δ sufficiently large, one concludes that T is representable by integration, and taking A such that kV k((K ∼ A) ∩ E(b)) is small
yields kT k(((Rn ∼ B) ∼ U) × J) = 0. The conclusion now follows from Theorem 9.12 (1) (3).
Remark 9.14. The conditions on ̺ are satisfied if, for instance, for any compact subset K of Rn ∼ B , there holds
Z
either
U ∩K
|V D f | dkV k < ∞,
lim (kV k x U ∩ K)(1) (V D f − V D fi ) = 0,
i→∞
lim sup (kV k x U ∩ K)(q) (V D fi ) < ∞ for some 1 < q ≤ ∞;
or
i→∞
in fact, if I is a compact subset of J and inf I > b ∈ J , then, in the first case,
lim sup
i→∞
Z
K∩A∩{x : fi (x)∈I}
|V D fi | dkV k ≤
Z
K∩A∩{x : f (x)>b}
|V D f | dkV k
whenever A is kV k measurable, and, in the second case,
̺(K, I, b, δ) ≤ δ1−1/q lim sup (kV k x U ∩ K)(q) (V D fi )
for 0 < δ < ∞.
i→∞
Lemma 9.15. If f is a nonnegative L 1 measurable R̄-valued function, O ⊂ R,
= 0, ε > 0, and j ∈ P , then there exist b1 , . . . , bj such that
L 1 (O)
ε(i − 1) < bi < εi and bi ∉ O
j
X
(bi − bi−1 )f (bi ) ≤ 2
i=1
for i = 1, . . . , j,
Z
f dL 1 ,
where b0 = 0.
Proof. It is sufficient to choose bi such that
ε(i − 1) < bi < εi,
bi ∉ O,
εf (bi ) ≤
Z εi
ε(i−1)
f dL 1
❐
for i = 1, . . . , j , and note bi − bi−1 ≤ 2ε.
The second application of the characterisation in Theorem 9.12 concerns the
multiplication of an element f of TG (V ) by a real-valued Lipschitzian function.
Its proof is based on a suitably chosen approximation of f by step functions.
Theorem 9.16. Suppose that m, n, U , V , G, and B are as in Definition 9.1,
f ∈ TG (V ), g : U → {y : 0 ≤ y < ∞}, and
Z
U ∩K
|f | + |V D f | dkV k < ∞,
Lip(g|K) < ∞,
whenever K is a compact subset of Rn ∼ B . Then, gf ∈ TG (V ).
1044
U LRICH M ENNE
Proof. Define h = gf , and note h ∈ T(V ) by Theorem 8.20 (4). Define a
function c by c = ((Rn ∼ B) × R) ∩ Clos g , and note that c is a function satisfying
dmn c = U ∪ G, c|U = g , and Lip(c|K) < ∞ whenever K is a compact subset of
Rn ∼ B . Moreover, let
J = {y : 0 < y < ∞},
A = (U × J) ∩ {(x, y) : h(x) > y},
and define p : (Rn ∼ B) × J → Rn ∼ B by p(x, y) = x for x ∈ Rn ∼ B and
y ∈ J . Noting (kV k+kδV k)(K ∩{x : h(x) > y}) < ∞ whenever K is a compact
subset of Rn ∼ B and y ∈ J , the proof may be carried out by showing that the
distribution T ∈ D ′ ((Rn ∼ B) × J, Rn ) defined by
Z T (ϕ) =
((δV ) x{x : h(x) > y})(ϕ(·, y)|U)
J
Z
−
P♮ • D ϕ(·, y)(x) dV (x, P ) dL 1 y
{x : h(x)>y}×G(n,m)
for ϕ ∈ D ((Rn ∼ B) × J, Rn) satisfies the conditions of Theorem 9.12 (3) with f
replaced by h.
For this purpose, define subsets D(y), E(υ), and F (y, υ) of U ∪ G, varifolds
Wy ∈ RV m (Rn ∼ B), and distributions S(y) ∈ D ′ (Rn ∼ B, Rn ) and Σ(y, υ) ∈
D ′ (Rn ∼ B, Rn ) by
D(y) = {x : f (x) > y},
E(υ) = {x : c(x) > υ},
Z
F (y, υ) = D(y) ∩ E(υ), Wy (k) =
k dV ,
D(y)×G(n,m)
Z
S(y)(θ) = ((δV ) x D(y))(θ|U) −
P♮ • D θ(x) dV (x, P ),
D(y)×G(n,m)
Z
Σ(y, υ)(θ) = ((δV ) x F (y, υ))(θ|U) −
P♮ • D θ(x) dV (x, P )
F(y,υ)×G(n,m)
n
whenever y, υ ∈ J , k ∈ K ((R ∼ B) × G(n, m)), and θ ∈ D (Rn ∼ B, Rn ). Let
O consist of all b ∈ J violating the following condition: S(b) is representable by
integration and kS(b)k((Rn ∼ B) ∼ U) = 0. Note L 1 (J ∼ O) = 0 by Definition
9.1, and
kδWb k is a Radon measure,
Σ(b, y) = S(b) x E(y) + Wb ∂E(y),
whenever b ∈ J ∼ O and y ∈ J . One readily verifies by means of Example 8.7 in
conjunction with [Fed69, 2.10.19 (4), 2.10.43] that c ∈ T(W ) and
W D c(x) = V D g(x)
for kV k almost all x ∈ D
whenever D is kV k measurable, W ∈ RV m (Rn ∼ B), W (k) =
Z
k ∈ K ((Rn ∼ B) × G(n, m)), and kδW k is a Radon measure.
D×G(n,m)
k dV for
Weakly Differentiable Functions on Varifolds
1045
Whenever N is a nonempty finite subset of J ∼ O , we now define functions
fN : dmn f → {y : 0 ≤ y < ∞} and hN : dmn f → {y : 0 ≤ y < ∞} by
fN (x) = sup({0} ∪ (N ∩ {y : x ∈ D(y)})),
hN (x) = fN (x)g(x),
whenever x ∈ dmn f , and also distributions ΘN ∈ D ′ ((Rn ∼ B) × J, Rn) and
TN ∈ D ′ ((Rn ∼ B) × J, Rn) by
ΘN (ϕ) =
Z ((δV ) x{x : fN (x) > y})(ϕ(·, y)|U)
Z
−
P♮ • D ϕ(·, y)(x) dV (x, P ) dL 1 y,
{x : fN (x)>y}×G(n,m)
Z TN (ϕ) =
((δV ) x{x : hN (x) > y})(ϕ(·, y)|U)
J
Z
−
P♮ • D ϕ(·, y)(x) dV (x, P ) dL 1 y
J
{x : hN (x)>y}×G(n,m)
whenever ϕ ∈ D ((Rn ∼ B) × J, Rn).
Next, it will be shown that if N is a nonempty finite subset of J ∼ O , then
kTN k(X × J) ≤ ((p# kΘN k) x X)(c) +
Z
U ∩X
f |V D g| dkV k
whenever X is an open subset of Rn ∼ B . For this purpose, suppose j ∈ P and
0 = b0 < b1 < · · · < bj < ∞ satisfy N = {bi : i = 1, . . . , j}, notice that
fN (x) = bi
fN (x) = 0
fN (x) = bj
if x ∈ D(bi ) ∼ D(bi+1) for some i = 1, . . . , j − 1,
if x ∈ (dmn f ) ∼ D(b1 ),
if x ∈ D(bj ),
and define distributions Ψ1 , Ψ2 ∈ D ′ ((Rn ∼ B) × J, Rn) by
Ψ1 (ϕ) =
Ψ2 (ϕ) =
Z J
Z j−
X1
J
y
b1
j X
y
y
+
S(bi ) x E
∼E
(ϕ(·, y)) dL 1 y,
bi
bi−1
i=2
S(b1 ) x E
i=1
(Wbi − Wbi+1 ) ∂E
y
bi
+ Wbj ∂E
y
bj
(ϕ(·, y)) dL 1 y
U LRICH M ENNE
1046
for ϕ ∈ D ((Rn ∼ B) × J, Rn ). One computes
ΘN (ϕ) =
j Zb
i
X
i=1
bi−1
for ϕ ∈ D ((Rn ∼ B) × J, Rn ),
S(bi )(ϕ(·, y)) dL 1 y
and deduces from Theorem 3.4 (2) with U replaced by Rn ∼ B that
kΘN k(d) =
j Zb
i
X
i=1
kS(bi )k(d(·, y)) dL 1 y
bi−1
whenever d is an R̄-valued kΘN k integrable function. Noting
y
y
{x : hN (x) > y} ∩ (D(bi ) ∼ D(bi+1 )) = F bi ,
∼ F bi+1 ,
,
bi
bi
{x : hN (x) > y} ∩ (U ∼ D(b1 )) = 0,
!
y
{x : hN (x) > y} ∩ D(bj ) = F bj ,
bj
for i = 1, . . . , j − 1, and y ∈ J , one obtains
TN (ϕ) =
Z j−
X1 y
y
y
− Σ bi+1 ,
+ Σ bj ,
(ϕ(·, y)) dL 1 y
Σ bi ,
b
b
b
J
i
i
j
i=1
whenever ϕ ∈ D ((Rn ∼ B) × J, Rn). If we use Remark 5.2 to compute
!
y
y
y
− Σ bi+1 ,
+ Σ bj ,
Σ bi ,
bi
bi
bj
i=1
X
j
j
X
y
y
S(bi ) x E
=
Wbi ∂E
+
b
b
i
i
i=1
i=1
j−1 X
j−1
−
X
S(bi+1 ) x E
i=1
= S(b1 ) x E
y
b1
j−1
+
X
i=1
(Wbi
y
bi
j−1
−
X
i=1
Wbi+1 ∂E
y
bi
y
y
S(bi ) x E
∼E
bi
bi−1
i=2
!
y
y
+ Wbj ∂E
− Wbi+1 ) ∂E
bi
bj
j
+
X
Weakly Differentiable Functions on Varifolds
1047
whenever y ∈ J , we obtain TN = Ψ1 + Ψ2 . Moreover, the quantity kΨ1 k(X × J)
does not exceed
Z X
j
y
y
y
kS(b1 )k x E
kS(bi )k x E
+
∼E
(X) dL 1 y
b1
b
b
J
i
i−
1
i=2
=
=
j Z
X
y
(kS(bi )k x X) E
dL 1 y
b
J
i
i=1
j Z
X
y
dL 1 y
(kS(bi )k x X) E
−
b
J
i−
1
i=2
j
X
(bi − bi−1 )(kS(bi )k x X)(c) = ((p# kΘN k) x X)(c),
i=1
and, by using Remark 8.5 and Theorem 8.30, the quantity kΨ2 k(X × J) may be
bounded by
Z j−
X1 (Wb − Wb ) ∂E y
i
i+1
b
J
i
i=1
bi
Z
bi
Z
j−1
=
X
i=1
j−1
=
X
i=1
=
Z
U ∩X
X
+ Wb ∂E y
j
b
j
(X) dL 1 y
|(Wbi − Wbi+1 ) D c| dkWbi − Wbi+1 k + bj
X∩(D(bi ) ∼ D(bi+1 ))
|V D g| dkV k + bj
Z
X
Z
X∩D(bj )
|Wbj D c| dkWbj k
|V D g| dkV k
f |V D g| dkV k.
Next, it will be proven that
kT k(X × J) ≤
Z
U ∩X
2g|V D f | + f |V D g| dkV k
whenever X is an open subset of Rn ∼ B . Recalling the formula for kΘN k, one may
use Lemma 9.15 with f (y) replaced by (kS(y)k x X)(c) for y ∈ J to construct
a sequence N(i) of nonempty finite subsets of J ∼ O such that
((p# kΘN(i) k) x X)(c) ≤ 2
Z
J
(kS(y)k x X)(c) dL 1 y,
dist(y, N(i)) → 0 as i → ∞, for y ∈ J.
Define A(i) = (U × J) ∩ {(x, y) : hN(i) (x) > y} for i ∈ P . Noting
hN(i) (x) → h(x)
as i → ∞, for x ∈ dmn f ,
U LRICH M ENNE
1048
and recalling (kV k + kδV k)(K ∩ {x : h(x) > y}) < ∞ for y ∈ J , whenever K is
a compact subset of Rn ∼ B , one infers
((kV k + kδV k) × L 1 )(C ∩ A ∼ A(i)) → 0
as i → ∞ whenever C is a compact subset of (Rn ∼ B) × J . Since A(i) ⊂ A for
i ∈ P , it follows by means of Fubini’s theorem that TN(i) → T as i → ∞, and, in
conjunction with the assertion of the preceding paragraph, we have
kT k(X × J) ≤ 2
Z
J
(kS(y)k x X)(c) dL 1 y +
Z
U ∩X
f |V D g| dkV k.
❐
Therefore, the assertion of the present paragraph is implied by (1), (4), and (6) of
Theorem 9.12.
Finally, the assertion of the preceding paragraph extends to all Borel subsets X
of Rn ∼ B by approximation, and the conclusion follows.
10. E MBEDDINGS INTO L EBESGUE S PACES
Introduction. In this section, a variety of Sobolev Poincaré type inequalities
for generalised weakly differentiable functions is established by means of the relative isoperimetric inequalities contained in Corollaries 7.9 and 7.11. The key is
constituted by local estimates under a smallness condition on set of points where
the nonnegative function is positive (see Theorem 10.1 (1)). These estimates are
formulated in such a way as to improve in the case where the function satisfies a
zero boundary value condition on an open part of the boundary. Consequently,
Sobolev inequalities are essentially a special case (see Theorem 10.1 (2)). Local
summability results also follow (see Corollary 10.3). Finally, versions without
the previously hypothesised smallness condition are derived in Theorems 10.7
and 10.9.
The differentiability results that will be derived in Theorems 11.2 and 11.4 are
based on Theorem 10.1 (1), whereas the oscillation estimates that will be proven
in Theorems 13.1 and 13.3 employ Theorem 10.9.
The formal pattern to derive Sobolev Poincaré type inequalities with zero median and Sobolev inequalities from isoperimetric inequalities is clearly known (see
Remark 10.2). Some care is needed, however, in verifying that all operations necessary have in fact been provided in Sections 8 and 9.
Theorem 10.1. Suppose 1 ≤ M < ∞. Then, there exists a positive, finite number Γ with the following property.
If m, n, p, U , V , and ψ are as in Hypothesis 6.1, n ≤ M , G is a relatively open
subset of Bdry U , B = (Bdry U) ∼ G, f ∈ TG (V ), and E = {x : f (x) > 0}, then
the following two statements hold:
(1) Suppose 1 ≤ Q ≤ M , 0 < r < ∞,
kV k(E) ≤ (Q − M −1 )α(m)r m ,
kV k(E ∩ {x : Θ m (kV k, x) < Q}) ≤ Γ −1 r m ,
Weakly Differentiable Functions on Varifolds
1049
and A = U ∩ {x : U(x, r ) ⊂ Rn ∼ B}. Then, the following four implications
hold:
(a) If p = 1, β = ∞ if m = 1 and β = m/(m − 1) if m > 1, then
(kV k x A)(β) (f ) ≤ Γ (kV k(1) (V D f ) + kδV k(f )).
(b) If p = m = 1 and ψ(E) ≤ Γ −1 , then
(kV k x A)(∞) (f ) ≤ Γ kV k(1) (V D f ).
(c) If 1 ≤ q < m = p and ψ(E) ≤ Γ −1 , then
(kV k x A)(mq/(m−q)) (f ) ≤ Γ (m − q)−1 kV k(q) (V D f ).
(d) If 1 < p = m < q ≤ ∞ and ψ(E) ≤ Γ −1 , then
(kV k x A)(∞) (f ) ≤ Γ 1/(1/m−1/q) kV k(E)1/m−1/q kV k(q) (V D f ).
(2) Suppose G = Bdry U and kV k({x : f (x) > y}) < ∞ for 0 < y < ∞.
Then, the following four implications hold:
(a) If p = 1, β = ∞ if m = 1 and β = m/(m − 1) if m > 1, then
kV k(β) (f ) ≤ Γ (kV k(1) (V D f ) + kδV k(f )).
(b) If p = m = 1 and ψ(E) ≤ Γ −1 , then
kV k(∞) (f ) ≤ Γ kV k(1) (V D f ).
(c) If 1 ≤ q < m = p and ψ(E) ≤ Γ −1 , then
kV k(mq/(m−q)) (f ) ≤ Γ (m − q)−1 kV k(q) (V D f ).
(d) If 1 < p = m < q ≤ ∞ and ψ(E) ≤ Γ −1 , then
kV k(∞) (f ) ≤ Γ 1/(1/m−1/q) kV k(E)1/m−1/q kV k(q) (V D f ),
where 0 · ∞ = ∞ · 0 = 0.
Proof. Denote by (1c)′ (respectively, (2c)′ ) the implication resulting from (1c)
(respectively, (2c)) through omission of the factor (m − q)−1 and addition of the
requirement q = 1. It is sufficient to construct functions Γ(1a) , Γ(1b) , Γ(1c) , Γ(1c)′ ,
Γ(1d) , Γ(2a) , Γ(2b) , Γ(2c) , Γ(2c)′ , and Γ(2d) corresponding to the implications (1a), (1b),
(1c), (1c)′ , (1d), (2a), (2b), (2c), (2c)′ , and (2d) whose value at M for 1 ≤ M < ∞
U LRICH M ENNE
1050
is a positive, finite number such that the respective implication is true for M with
Γ replaced by this value. Define
Γ(1a) (M) = Γ(1b) (M) = Γ(1c)′ (M) = Γ7.8 (M),
Γ(2a) (M) = Γ(1a) (sup{2, M}),
Γ(2b) (M) = Γ(1b) (sup{2, M}),
Γ(2c)′ (M) = Γ(1c)′ (sup{2, M}),
Γ(2c) (M) = M 2 Γ(2c)′ (M),
1
∆1 (M) = sup
,1 −
2
∆2 (M) =
2M
1
2M 2 − 1
1/M
,
,
1 − ∆1 (M)
∆3 (M) = M sup{α(m) : M ≥ m ∈ P },
inf{α(m) : M ≥ m ∈ P }
∆4 (M) =
,
2
∆5 (M) = sup{1, Γ(2c) (M)(∆2 (M)∆3 (M) + 1)},
∆6 (M) = 4M+1 ∆3 (M)M ∆5 (M)M+1 ,
Γ(1c) (M) = sup{2Γ(1c)′ (2M), ∆6 (M)(1 + ∆3 (M)Γ(1c)′ (2M))},
∆7 (M) = sup{1, Γ(1c)′ (2M)},
−1
M
M 1
,
∆8 (M) = inf inf{1, ∆3 (M) ∆4 (M)∆1 (M) }(1 − ∆1 (M)) ,
2
−1
∆9 (M) = 2∆7 (M)∆8 (M)
,
Γ(1d) (M) = sup{∆9 (M), ∆1 (M)−M Γ(1c)′ (2M)},
Γ(2d) (M) = Γ(1d) (sup{2, M}),
whenever 1 ≤ M < ∞.
In order to verify the asserted properties, suppose 1 ≤ M < ∞, and abbreviate δi = ∆i (M) whenever i = 1, . . . , 9. In the verification of assertion (X )
(respectively, (X )′ ), where X is one of 1a, 1b, 1c, 1d, 2a, 2b, 2c, and 2d (respectively, 1c and 2c), it will be assumed that the quantities occurring in its hypotheses
are defined and satisfy these hypotheses with Γ replaced by Γ(X) (M) (respectively,
Γ(X)′ (M)).
Step 1. Reduction to the case kV k(E) < ∞. Considering sup{f (x)−ε, 0} for ε > 0
instead of f (x) in (2) yields the asserted reduction by Lemma 8.12, Example
8.13 (4), and Lemma 9.9.
Step 2. Verification of the property of Γ(1a) . Define E(b) = {x : f (x) > b} for
0 ≤ b < ∞. If m = 1, then Γ7.8 (M)−1 ≤ kV ∂E(b)k(U) + kδV k(E(b)) for L 1
almost all b with 0 < b < (kV k x A)(β) (f ) by Corollary 7.9, and the conclusion
Weakly Differentiable Functions on Varifolds
1051
follows from Theorem 8.30. If m > 1, define
fb = inf{f , b},
g(b) = (kV k x A)(β) (fb ) ≤ bkV k(E)1/β < ∞
for 0 ≤ b < ∞, and use Minkowski’s inequality to conclude
0 ≤ g(b + y) − g(b) ≤ (kV k x A)(β) (fb+y − fb ) ≤ ykV k(A ∩ E(b))1/β
for 0 ≤ b < ∞ and 0 < y < ∞. Therefore, g is Lipschitzian, and one infers from
[Fed69, 2.9.19] and Corollary 7.9 that
0 ≤ g ′ (b) ≤ kV k(A ∩ E(b))1−1/m ≤ Γ7.8 (M)(kV ∂E(b)k(U) + kδV k(E(b)))
for L 1 almost all 0 < b < ∞; thus, (kV k x A)(β) (f ) = limb→∞ g(b) =
Z∞
0
g ′ dL 1
by [Fed69, 2.9.20], and Theorem 8.30 implies the conclusion.
Step 3. Verification of the properties of Γ(1b) and Γ(1c)′ . By omitting the terms involving δV from the proof of Step 2 and using Corollary 7.11 instead of Corollary 7.9,
a proof of Step 3 results.
Step 4. Verification of the properties of Γ(2a) , Γ(2b) , and Γ(2c)′ . Taking Q = 1, one
may apply (1a), (1b), and (1c)′ with M replaced by sup{2, M} and a sufficiently
large number r .
Step 5. Verification of the property of Γ(2c) . One may assume f to be bounded by
Lemma 8.12 and Example 8.13 (4). Noting that f q(m−1)/(m−q) ∈ TBdry U (V ) by
Lemma 9.9, one now applies (2c)′ with f replaced by f q(m−1)/(m−q) to deduce
the assertion by the method of [Fed69, 4.5.15].
Step 6. Verification of the property of Γ(1c) . One may assume kV k(q) (V D f ) < ∞,
and, possibly using homotheties, also r = 1. Moreover, one may assume f to be
bounded by Lemma 8.12 and Example 8.13 (4); hence,
Z
|f | + |V D f | dkV k < ∞
by Theorem 8.20 (1) and Hölder’s inequality. Define
ri = δ1 +
(i − 1)(1 − δ1 )
,
M
Ai = U ∩ {x : U(x, ri ) ⊂ Rn ∼ B}
whenever i = 1, . . . , m. Observe that
Q − M −1 ≤ r1m (Q − (2M)−1 ),
ri+1 − ri =
2
δ2
r1m ≥
1
,
2
rm ≤ 1,
for i = 1, . . . , m − 1.
U LRICH M ENNE
1052
Choose gi ∈ E (U, R) with
spt gi ⊂ Ai , gi (x) = 1 for x ∈ Ai+1 ,
0 ≤ gi (x) ≤ 1 for x ∈ U, Lip gi ≤ δ2 ,
whenever i = 1, . . . , m − 1. Notice that gi f ∈ T(V ) by Theorem 8.20 (4) with
V D (gi f )(x) = V D gi (x)f (x) + gi (x)V D f (x)
for kV k almost all x.
Moreover, one infers gi f ∈ TBdry U (V ) by Theorem 9.16 and Lemma 9.4.
If i ∈ {1, . . . , m − 1} and 1 − (i − 1)/m ≥ 1/q ≥ 1 − i/m, then by (2c) and
Hölder’s inequality,
(kV k x Ai+1 )(mq/(m−q)) (f ) ≤ kV k(mq/(m−q)) (gi f )
≤ Γ(2c) (M)(m − q)−1 (δ2 (kV k x Ai )(q) (f ) + kV k(q) (V D f ))
≤ δ5 (m − q)−1 ((kV k x Ai )(m/(m−i)) (f ) + kV k(q) (V D f )).
In particular, replacing q by m/(m − i) and using Hölder’s inequality in conjunction with Theorem 8.20 (1), one infers
(kV k x Ai+1 )(m/(m−i−1)) (f )
≤ 2δ3 δ5 ((kV k x Ai )(m/(m−i)) (f ) + kV k(q) (V D f ))
whenever i ∈ {1, . . . , m − 2} and 1 − i/m ≥ 1/q; by choosing j ∈ {1, . . . , m − 1}
with 1 − (j − 1)/m ≥ 1/q > 1 − j/m and iterating this j − 1 times, one also
obtains
(kV k x Aj )(m/(m−j)) (f )
≤ (4δ3 δ5 )j−1 ((kV k x A1 )(m/(m−1)) (f ) + kV k(q) (V D f )).
Together, this yields
(kV k x Aj+1 )(mq/(m−q)) (f )
≤ δ6 (m − q)−1 ((kV k x A1 )(m/(m−1)) (f ) + kV k(q) (V D f )),
and the conclusion then follows from (1c)′ with M and r replaced by 2M and r1 .
Step 7. Verification of the property of Γ(1d) . Assume
r = 1,
kV k(E) > 0,
kV k(q) (V D f ) < ∞,
abbreviate β = m/(m − 1) and α = 1/(1/m − 1/q), and suppose
λ = kV k(E),
kV k(q) (V D f ) < κ < ∞.
Weakly Differentiable Functions on Varifolds
1053
Notice that 0 < δ1 < 1, 0 < δ8 < 1, α ≥ 1 and λ > 0, and define
ri = 1 − (1 − δ1 )i ,
1/m−1/q
si = δ α
κ(1 − 2−i ),
9λ
Xi = Rn ∩ {x : dist(x, B) > ri },
Ui = U ∩ Xi ,
Hi = Xi ∩ Bdry U,
Ci = (Bdry Ui ) ∼ Hi ,
Ui ×G(n,m)
Wi = V |2
,
ti = kV k(Ui ∩ {x : f (x) > si }),
fi (x) = sup{f (x) − si , 0}
whenever x ∈ dmn f ,
whenever i is a nonnegative integer. Notice that
ri+1 − ri = δ1 (1 − δ1 )i ,
Ui+1 ⊂ Ui ,
1/m−1/q
κ 2−i−1 > 0
si+1 − si = δα
9λ
whenever i is a nonnegative integer. The conclusion is readily deduced from the
assertion
ti ≤ λδαi
whenever i is a nonnegative integer,
8
which will be proven by induction.
The case i = 0 is trivial. To prove the case i = 1, one applies (1c)′ (with
M and r replaced by 2M and δ1 ) and Hölder’s inequality in conjunction with
Theorem 8.20 (1) to obtain
(kV k x U1 )(β) (f ) ≤ Γ(1c)′ (2M)kV k(1) (V D f ) ≤ δ7 λ1−1/q κ ;
hence,
1−1/m
t1
≤ (s1 − s0 )−1 (kV k x U1 )(β) (f )
α(1−1/m)
1−1/m
2δ7 ≤ λ1−1/m δ8
≤ δ−α
9 λ
.
Assuming the assertion to be true for some i ∈ P , notice that
λ ≤ δ3 ,
αi ≥ 1,
m
im
ti ≤ λδαi
≤
8 ≤ δ4 δ1 (1 − δ1 )
fi ∈ TG (V ),
Hi = Xi ∩ Bdry Ui ,
1
α(m)(ri+1 − ri )m ,
2
fi |Xi ∈ THi (Wi ),
Ui+1 ⊂ Ui ∩ {x : U(x, ri+1 − ri ) ⊂ Rn ∼ Ci },
by Lemma 8.12, Example 8.13 (4) and Lemma 9.5, and Remark 9.6 with f , X ,
H , W , C , and r replaced by fi , Xi , Hi , Wi , Ci , and ri+1 − ri . Therefore, (1c)′
applied with U , V , Q, M , G, B , f , and r replaced by Ui , Xi , 1, 2M , Hi , Ci , fi |Xi ,
and ri+1 − ri yields
(kV k x Ui+1 )(β) (fi ) ≤ Γ(1c)′ (2M)(kV k x Ui )(1) (V D fi );
U LRICH M ENNE
1054
hence, by using Hölder’s inequality in conjunction with Lemma 8.12, Example 8.13 (4), and Theorem 8.20 (1), we obtain
1−1/q
(kV k x Ui+1 )(β) (fi ) ≤ δ7 ti
αi(1−1/q)
κ ≤ δ7 λ1−1/q δ8
κ.
1−1/q)
α(1−1/m)
α(1−1/m)
Noting δ−α
and 2δα(
≤ δ8
, it follows that
9 2δ7 ≤ δ8
8
1−1/m
ti+1
≤ (si+1 − si )−1 (kV k x Ui+1 )(β) (fi )
α(1−1/q) i
≤ λ1−1/m δ−α
9 2δ7 (2δ8
1−1/m α(1−1/m)(i+1)
≤λ
δ8
)
,
❐
and the assertion is proven.
Step 8. Verification of the property of Γ(2d) . Taking Q = 1, one may apply (1d) with
M replaced by sup{2, M} and a sufficiently large number r .
Remark 10.2. The role of Corollaries 7.9 and 7.11, respectively, in Step 2
and Step 3 of the preceding proof is identical to the one of Allard [All72, 7.1] in
Allard [All72, 7.3].
The method of deduction of (2c) from (2c)′ is classical (see [Fed69, 4.5.15]).
In the context of Lipschitzian functions, the method of deduction of (1c) from
(1c)′ and (2c) is outlined by Hutchinson in [Hut90, pp. 59–60].
The iteration procedure employed in the proof (1d) bears formal resemblance
to that of Stampacchia [Sta66, Lemma 4.1 (i)]. However, here the estimate of ti+1
in terms of ti requires a smallness hypothesis on ti .
Qualitative embedding results now follow without particular difficulty.
Corollary 10.3. Suppose that m, n, U , V , and p are as in Hypothesis 6.1,
1 ≤ q ≤ ∞, Y is a finite-dimensional normed vector space, f ∈ T(V , Y ), and
n
V D f ∈ Lloc
q (kV k, Hom(R , Y )).
Then, the following four statements hold:
loc
(1) If m > 1 and f ∈ Lloc
1 (kδV k, Y ), then f ∈ Lm/(m−1) (kV k, Y ).
(2) If m = 1, then f ∈ Lloc
∞ (kV k + kδV k, Y ).
(3) If 1 ≤ q < m = p, then f ∈ Lloc
mq/(m−q) (kV k, Y ).
(4) If 1 < m = p < q ≤ ∞, then f ∈ Lloc
∞ (kV k, Y ).
Proof. In view of Remark 8.16 and Remark 9.2, it is sufficient to consider the
case Y = R and f ∈ T0 (V ). Assume p = 1 in case of (1) or (2), and define ψ as
in Hypothesis 6.1. Moreover, assume (kV k + ψ)(U) + kV k(q) (V D f ) < ∞, and,
in case of (1), also kδV k(1) (f ) < ∞. Suppose K is a compact subset of U . Choose
0 < r < ∞ with U(x, r ) ⊂ U for x ∈ K and 0 < s < ∞ with
kV k(E) ≤
1
α(m)r m ,
2
ψ(E) ≤ Γ10.1 (sup{2, n})−1 ,
Weakly Differentiable Functions on Varifolds
1055
where E = {x : f (x) > s}. Select g ∈ E (R, R) with g(t) = 0 if t ≤ s and
g(t) = t if t ≥ s + 1, and notice that g ◦ f ∈ T0 (V ) and
kV D (g ◦ f )(x)k ≤ (Lip g)kV D f k(x)
for kV k almost all x
❐
by Lemma 8.12 with Υ = {t : t ≤ s} and Remark 9.2. Applying Theorem 10.1 (1)
with M , G, f , and Q replaced by sup{2, n}, 0, g ◦ f , and 1, and, in case of (2),
noting Corollary 8.33, we see that the conclusion follows.
Next, we carry out preparations for the treatment of Sobolev Poincaré embedding results involving several means. The first is an elementary inequality.
Lemma 10.4. If 1 ≤ q ≤ p ≤ ∞, q < ∞, µ measures X , G is a countable
collection of µ measurable {t : 0 ≤ t ≤ ∞}-valued functions, 1 ≤ κ < ∞,
card(G ∩ {g : g(x) > 0}) ≤ κ
and f (x) =
P
g∈G
for µ almost all x,
g(x) for µ almost all x , then
X
1/q
µ(p) (f ) ≤ κ
µ(p) (g)q
,
g∈G
µ(∞) (f ) ≤ κ sup({0} ∪ {µ(∞) (g) : g ∈ G}).
P
Proof. Abbreviating h(x) = g∈G g(x)q for µ almost all x , one estimates
q/p
X Z
X
≤ κq
g p dµ
µ(p) (g)q if p < ∞,
µ(p) (f )q ≤ κ q
g∈G
q
q
µ(∞) (f ) ≤ κ µ(∞) (h) ≤ κ
q
X
g∈G
µ(∞) (g)q ,
❐
g∈G
and the conclusion follows.
The following is a variant of Federer’s version of a “Whitney partition of unity”
(see [Fed69, 3.1.13]), adapted to finite-dimensional normed vector spaces.
Lemma 10.5. Suppose Y is aSfinite-dimensional normed vector space, Φ is a
family of open subsets of Y , and h : Φ → {t : 0 < t < ∞} satisfies
[
1
sup{inf{1, dist(y, Y ∼ Υ)} : Υ ∈ Φ} whenever y ∈ Φ.
20
S
Then, there exists a countable subset B of Φ, and, for each b ∈ B , an associated
function gb : Y → R with the following properties:
(1) If y ∈ B , then By = B ∩ {b : B(b, 10h(b)) ∩ B(y, 10h(y)) ≠ 0} satisfies
h(y) =
card By ≤ (129)dim Y .
(2) If b ∈ B , then 0 ≤ gb (y) ≤ 1 for y ∈ Y , spt gb ⊂ B(b, 10h(b)), and
Lip(gb |B(y, 10h(y))) ≤ (130)dim Y h(y)−1
(3) If y ∈ Y , then
P
b∈B
gb (y) = 1.
whenever y ∈
[
Φ.
U LRICH M ENNE
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Proof. Assume dim Y > 0 and observe that [Fed69, 3.1.12] remains valid with
Rm and m replaced by Y and dim Y ; in fact, one modifies the proof by choosing
a nonzero translation-invariant Radon measure ν over Y and replacing α(m) by
ν B(0, 1). Then, one modifies the proof of [Fed69, 3.1.13] by taking B to be the
set named S there, γ(t) = sup{0, inf{1, 2 − t}} for t ∈ R (hence Lip µ = 1), and
adding the estimates
Lip ub ≤ h(y)−1 ,
Lip(σ |B(y, 10h(y))) ≤ (129)dim Y h(y)−1 ,
Lip(vb |B(y, 10h(y))) ≤ Lip ub + Lip(σ |B(y, 10h(y)))
whenever y ∈
S
Φ and b ∈ By ; hence, one may take gb = vb for b ∈ B .
❐
≤ (130)dim Y h(y)−1
Lemma 10.6. Suppose Y is a finite-dimensional normed vector space, D is a
closed subset of Y , and 0 < s < ∞.
Then, there exists a countable family G with the following properties:
(1) If g ∈ G, then g : Y → R and spt g ⊂ U(d, s) for some d ∈ D .
dim Y .
(2) If y ∈ Y , then card(GP∩ {g : B(y, s/4) ∩ spt
Pg ≠ 0}) ≤ (129)
(3) There holds D ⊂ Int{ g∈G g(y) = 1} and g∈G g(y) ≤ 1 for y ∈ Y .
(4) If g ∈ G, then 0 ≤ g(y) ≤ 1 for y ∈ Y and Lip g ≤ 40 · (130)dim Y s −1.
❐
Proof. Assume s = 1. Defining Φ = {Y ∼ D} ∪ {U(d, 1) : d ∈ D}, observe
1
that the function h resulting from Lemma 10.5 satisfies h(y) ≥ 40
for y ∈ Y ;
hence, we verify that we can take G = {gb : D ∩ spt gb ≠ 0}.
Now, Sobolev Poincaré-type inequalities with several means may be derived
by employing a localisation procedure in the target space.
Theorem 10.7. Suppose 1 ≤ M < ∞. Then, there exists a positive, finite number Γ with the following property. If m, n, p, U , V , and ψ are as in Hypothesis 6.1,
Y is a finite-dimensional normed vector space, sup{dim Y , n} ≤ M , f ∈ T(V , Y ),
1 ≤ Q ≤ M , N ∈ P , 0 < r < ∞,
kV k(U) ≤ (Q − M −1 )(N + 1)α(m)r m ,
kV k({x : Θ m (kV k, x) < Q}) ≤ Γ −1 r m ,
A = {x : U(x, r ) ⊂ U}, and fΥ : dmn f → R is defined by
fΥ (x) = dist(f (x), Υ)
whenever x ∈ dmn f , 0 ≠ Υ ⊂ Y ,
then the following four statements hold:
Weakly Differentiable Functions on Varifolds
1057
(1) If p = 1, β = ∞ if m = 1 and β = m/(m − 1) if m > 1, then there exists
a subset Υ of Y such that 1 ≤ card Υ ≤ N and
(kV k x A)(β) (fΥ ) ≤ Γ N 1/β (kV k(1) (V D f ) + kδV k(fΥ )).
(2) If p = m = 1 and ψ(U) ≤ Γ −1 , then there exists a subset Υ of Y such that
1 ≤ card Υ ≤ N and
(kV k x A)(∞) (fΥ ) ≤ Γ kV k(1) (V D f ).
(3) If 1 ≤ q < m = p and ψ(U) ≤ Γ −1 , then there exists a subset Υ of Y such
that 1 ≤ card Υ ≤ N and
(kV k x A)(mq/(m−q)) (fΥ ) ≤ Γ N 1/q−1/m (m − q)−1 kV k(q) (V D f ).
(4) If 1 < p = m < q ≤ ∞ and ψ(U) ≤ Γ −1 , then there exists a subset Υ of Y
such that 1 ≤ card Υ ≤ N and
(kV k x A)(∞) (fΥ ) ≤ Γ 1/(1/m−1/q) r 1−m/q kV k(q) (V D f ).
Proof. Define
∆3 = 40 · (130)M ,
1
1
,1 −
∆2 = sup
2
2M 2 − 1
∆4 = ∆−M
2 Γ10.1 (2M),
∆7 = sup{α(m) : M ≥ m ∈ P },
∆8 = 4M∆6 ∆7 + Γ10.1 (2M),
M
∆1 = sup{1, Γ7.8 (M)} ,
∆5 =
1/M
,
1
inf{α(m) : M ≥ m ∈ P }, ∆6 = sup{1, Γ10.1 (2M)}
2
1
−M
· (∆33 + 1)∆−
,
5 (1 − ∆2 )
∆9 = M∆7 Γ10.1 (2M),
Γ = sup{∆1 , ∆4 , ∆8 , 2∆6 (1 + ∆9 )},
and notice that ∆5 ≤ 1 ≤ inf{∆6 , ∆7 }.
In order to verify that Γ has the asserted property, suppose m, n, p, U , V , ψ,
Y , f , Q, N , r , A, and fΥ are related to Γ as in the body of the theorem. Abbreviate
ι = mq/(m − q) in case of (3) and α = 1/m − 1/q in case of (4). Moreover,
define
κ = kV k(1) (V D f )
−1
κ = (m − q)
kV k(q) (V D f )
1/α
κ = ∆9 kV k(q) (V D f )
and assume r = 1 and dmn f = U .
in case of (1) or (2),
in case of (3),
in case of (4),
U LRICH M ENNE
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First, the case κ = 0 will be considered. For this purpose, choose Ξ and υ as
in Theorem 8.34, and let
Π = Ξ ∩ {W : kW k(U) ≤ (Q − M −1 )α(m)}.
Applying Corollaries 7.9 and 7.11 with V , E , and B replaced by W , U , and Bdry U ,
one infers
kW k(A)1/β ≤ ∆1 kδW k(U)
in case of (1), where 00 = 0,
in case of (2) or (3) or (4)
kW k(A) = 0
whenever W ∈ Π. Since card(Ξ ∼ Π) ≤ N , there exists Υ such that
υ[Ξ ∼ Π] ⊂ Υ ⊂ Y
and 1 ≤ card Υ ≤ N.
In case of (1), it follows that, by using Remark 6.10,
(kV k x A)(β) (fΥ ) ≤
X
W ∈Π
≤ ∆1
dist(υ(W ), Υ)kW k(A)1/β
X
W ∈Π
dist(υ(W ), Υ)kδW k(U)
= ∆1 kδV k(fΥ ).
In the case of (2) or (3) or (4), the corresponding estimate is trivial.
Second, the case κ > 0 will be considered in the remaining proof. For this
purpose, assume κ < ∞, and define
X = {x : B(x, ∆2 ) ⊂ U},
B = Y ∩ {y : kV k(f
Choose Υ ⊂ Y satisfying
1 ≤ card Υ ≤ N
−1
s = ∆6 κ,
[U(y, s)]) > (Q − M −1 )α(m)}.
B ⊂ {y : dist(y, Υ) < 2s};
in fact, if B ≠ 0 then one may take Υ to be a maximal subset of B with respect
to inclusion such that {U(y, s) : y ∈ Υ} is disjointed. Abbreviate γ = dist(·, Υ),
and, in case of (1), define λ = κ + kδV k(γ ◦ f ) and assume λ < ∞. In case of (2)
or (3) or (4), define λ = κ . Let
D = Y ∩ {y : γ(y) ≥ 2∆6 λ},
E = f −1 [D].
Next, the following assertion will be shown:
1
α(m)(1 − ∆2 )m
2
kV k(X ∩ E) = 0
kV k(X ∩ E) ≤
in case of (1) or (3),
in case of (2) or (4).
Weakly Differentiable Functions on Varifolds
1059
To prove this, choose G as in Lemma 10.6, and let Eg = {x : g(f (x)) > 0}
for g ∈ G. Since D ∩ B = 0 as ∆6 λ ≥ s , it follows that
kV k(Eg ) ≤ (Q − M −1 )α(m) ≤ (Q − (2M)−1 )α(m)∆m
2 ,
1
−1 m
kV k(Eg ∩ {x : Θ m (kV k, x) < Q}) ≤ Γ −1 ≤ ∆−
4 ≤ Γ10.1 (2M) ∆2 ,
ψ(Eg ) ≤ Γ −1 ≤ Γ10.1 (2M)−1
in case of (2) or (3) or (4)
whenever g ∈ G. Denoting by cg the characteristic function of Eg , applying
Lemma 8.15 with Υ replaced by spt g , and noting Theorem 8.20 (1) and Remark 9.2, yields g ◦ f ∈ T0 (V ) with
|V D (g ◦ f )(x)| ≤ ∆3 s −1 cg (x)kV D f (x)k
for kV k almost all x
whenever g ∈ G. P
Moreover, denoting the
P function whose domain is U and whose
value at x equals g∈G (g ◦ f )(x) by g∈G g ◦ f , notice that
card(G ∩ {g : x ∈ Eg }) ≤ ∆3
for x ∈ U,
X
g∈G
{y : g(y) > 0} ⊂ {y : γ(y) ≥ ∆6 λ}
g ◦ f |E = 1,
for g ∈ G.
In the case of (1), one estimates, using Theorem 10.1 (1a) with M , G, f , and r
replaced by 2M , 0, g ◦ f , and ∆2 , that
X
X
kV k(X ∩ E)1/β ≤ (kV k x X)(β)
g◦f ≤
(kV k x X)(β) (g ◦ f )
g∈G
g∈G
X
≤ Γ10.1 (2M)
(∆3 s −1 (kV k x Eg )(1) (V D f ) + kδV k(g ◦ f ))
g∈G
1 2
≤ Γ10.1 (2M)(∆−
6 ∆3 + kδV k({x : γ(f (x)) ≥ ∆6 λ}))
1
M
≤ Γ10.1 (2M)(∆23 + 1)∆−
6 ≤ ∆5 (1 − ∆2 )
1/β
1
≤
,
α(m)(1 − ∆2 )m
2
where 00 = 0. In case of (2), using Theorem 10.1 (1b) with M , G, f , and r
replaced by 2M , 0, g ◦ f , and ∆2 , one estimates
(kV k x X)(∞)
X
g∈G
≤ Γ10.1 (2M)
X
g◦f ≤
(kV k x X)(∞) (g ◦ f )
X
g∈G
g∈G
∆3 s
−1
(kV k x Eg )(1) (V D f )
M
1
≤ Γ10.1 (2M)∆23 ∆−
6 ≤ ∆5 (1 − ∆2 ) < 1,
U LRICH M ENNE
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hence kV k(X ∩ E) = 0. In the case of (3), using Lemma 10.4 and Theorem
10.1 (1c) with M , G, f , and r replaced by 2M , 0, g ◦ f , and ∆2 , one estimates
kV k(X ∩ E)1/ι ≤ (kV k x X)(ι)
X
g◦f
g∈G
≤ ∆3
X
(kV k x X)(ι) (g ◦ f )q
g∈G
≤ Γ10.1 (2M)(m − q)−1 ∆23 s −1
1/q
X
(kV k x Eg )(q) (V D f )q
g∈G
M
1
≤ Γ10.1 (2M)∆33 ∆−
6 ≤ ∆5 (1 − ∆2 ) ≤
1/q
1
α(m)(1 − ∆2 )m
2
1/ι
.
In the case of (4), using Theorem 10.1 (1d) with M , G, f , and r replaced by 2M ,
1/α
0, g ◦ f , and ∆2 , and noting Γ10.1 (2M)1/α (M∆7 )α ≤ ∆9 , one obtains
1/α
(kV k x X)(∞) (g ◦ f ) ≤ ∆9 ∆3 s −1 (kV k x Eg )(q) (V D f )
for g ∈ G,
and therefore kV k(X ∩ E) = 0, since if q < ∞, then, using Lemma 10.4, we have
(kV k x X)(∞)
X
g∈G
1/α
X
1/q
g ◦ f ≤ ∆3
(kV k x X)(∞) (g ◦ f )q
≤ ∆9 ∆23 s −1
g∈G
X
(kV k x Eg )(q) (V D f )q
g∈G
1/q
1
∆33 ∆−
6
≤ ∆5 (1 − ∆2 )M < 1,
P
and if q = ∞, then (kV k x X)(∞) ( g∈G g ◦ f ) < 1 follows similarly.
≤
In case of (2) or (4), noting 2∆6 ∆19/α ≤ Γ 1/α in case of (4), the conclusion is
evident from the assertion of the preceding paragraph. Now, consider the case of
(1) or (3), and define h : Y → R by
h(y) = sup{0, γ(y) − 2∆6 λ}
for y ∈ Y .
Notice that h ◦ f ∈ T0 (V |2X×G(n,m) ) with
|V D (h ◦ f )(x)| ≤ kV D f (x)k
for kV k almost all x
by Lemma 8.15 with Υ replaced by Y and Remark 9.2. In the case of (1), applying Theorem 10.1 (1a) with M , U , V , G, f , Q, and r replaced by 2M , X ,
V |2X×G(n,m) , 0, h ◦ f |X , 1, and 1 − ∆2 implies
(kV k x A)(β) (h ◦ f ) ≤ Γ10.1 (2M)(kV k(1) (V D f ) + kδV k(h ◦ f )),
(kV k x A)(β) (γ ◦ f ) ≤ 4M∆6 ∆7 N 1/β λ + Γ10.1 (2M)λ ≤ Γ N 1/β λ
Weakly Differentiable Functions on Varifolds
1061
since kV k(A)1/β ≤ 2M∆7 N 1/β . In the case of (3), applying Theorem 10.1 (1c)
with M , U , V , G, f , Q, and r replaced by 2M , X , V |2X×G(n,m) , 0, h ◦ f |X , 1,
and 1 − ∆2 implies
(kV k x A)(ι) (γ ◦ f ) ≤ 4M∆6 ∆7 N 1/ι κ + Γ10.1 (2M)κ ≤ Γ N 1/ι κ,
since kV k(A)1/ι ≤ 2M∆7N 1/ι .
❐
(kV k x A)(ι) (h ◦ f ) ≤ Γ10.1 (2M)κ,
Remark 10.8. Comparing the preceding theorem to previous Sobolev Poincaré inequalities of the author (see [Men10, 4.4–4.6]), which treated the particular
case of f being the orthogonal projection onto an (n − m)-dimensional plane in
the context of integral varifolds, the present approach is significantly more general.
Yet, for the case of orthogonal projections, the previous results give somewhat
more precise information, as they include, for instance, the intermediate cases
1 < p < m as well as Lorentz spaces. Evidently, the question arises whether the
present theorem can be correspondingly extended.
In the case where the target space is one dimensional, somewhat sharper estimates may be derived in a much simpler fashion.
Theorem 10.9. Suppose m, n, p, U , V , and ψ are as in Hypothesis 6.1, n ≤
M < ∞, Λ = Γ10.1 (M), N ∈ P , 1 ≤ Q ≤ M , f ∈ T(V ), 0 < r < ∞, X is a finite
subset of U ,
kV k(U) ≤ (Q − M −1 )(N + 1)α(m)r m ,
kV k({x : Θ m (kV k, x) < Q}) ≤ Λ−1 r m ,
and A = {x : U(x, r ) ⊂ U}. Then, there exists a subset Υ of R with 1 ≤ card Υ ≤
N + card X such that the following four statements hold with g = dist(·, Υ) ◦ f :
(1) If p = 1, β = ∞ if m = 1 and β = m/(m − 1) if m > 1, then
(kV k x A)(β) (g) ≤ Λ(kV k(1) (V D f ) + kδV k(g)).
(2) If p = m = 1 and ψ(U ∼ X) ≤ Λ−1 , then
(kV k x A)(∞) (g) ≤ ΛkV k(1) (V D f ).
(3) If 1 ≤ q < m = p and ψ(U) ≤ Λ−1 , then
(kV k x A)(mq/(m−q)) (g) ≤ Λ(m − q)−1 kV k(q) (V D f ).
(4) If 1 < p = m < q ≤ ∞ and ψ(U) ≤ Λ−1 , then
(kV k x A)(∞) (g) ≤ Γ 1/(1/m−1/q) Q1/m−1/q r 1−m/q kV k(q) (V D f ),
where Γ = Λ sup{α(i) : M ≥ i ∈ P }.
U LRICH M ENNE
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Proof. Choose a nonempty subset Υ of R satisfying
card Υ ≤ N + card X,
kV k(U ∩ {x : f (x) ∈ I}) ≤ (Q − M
f [X] ⊂ Υ,
−1
)α(m)r m
for I ∈ Φ,
where Φ is the family of connected components of R ∼ Υ . Notice that
dist(b, Υ) = dist(b, R ∼ I) and
dist(b, R ∼ J) = 0
whenever b ∈ I ∈ Φ and I ≠ J ∈ Φ, and
dist(b, Υ) = dist(b, R ∼ I) = 0 whenever b ∈ Υ and I ∈ Φ.
Defining fI = dist(·, R ∼ I) ◦ f , one infers fI ∈ T0 (V ) and
|V D f (x)| = |V D fI (x)|
for kV k almost all x ∈ f −1[I],
V D f (x) = 0
for kV k almost all x ∈ f −1[Υ],
V D fI (x) = 0
for kV k almost all x ∈ f −1[R ∼ I]
whenever I ∈ Φ by Lemma 8.12, Example 8.13, Theorem 8.20 (1), and Remark 9.2.
The conclusion now will be obtained by employing Lemma 10.4 and applying
Theorem 10.1 (1) with G and f replaced by 0 and fI for I ∈ Φ. For instance, in
the case of (3), one estimates
(kV k x A)(mq/(m−q)) (g) ≤
X
I∈Φ
(kV k x A)(mq/(m−q)) (fI )q
X
1/q
≤λ
kV k(q) (V D fI )q
= λkV k(q) (V D f ),
1/q
I∈Φ
❐
where λ = Λ(m − q)−1 . The cases (1) and (2) then follow similarly. Defining
2
2
∆ = sup{α(i) : M ≥ i ∈ P }, we have that α(m)(1/m−1/q) ≤ ∆(1/m−1/q) ≤ ∆,
since ∆ ≥ 1. Therefore, also (4) follows similarly.
Remark 10.10. The method of deduction of Theorem 10.9 from Theorem 10.1 is that of Hutchinson [Hut90, Theorem 3], which is derived from
Hutchinson [Hut90, Theorem 1].
Remark 10.11. A nonempty choice of X will occur in Theorem 13.1.
Having established a solid base, the treatment of the further properties of generalised weakly differentiable functions and the two applications in the remaining
five sections is now comparably straightforward.
Weakly Differentiable Functions on Varifolds
1063
11. D IFFERENTIABILITY P ROPERTIES
Introduction. In this section, approximate differentiability (see also Theorem 11.2) and differentiability in Lebesgue spaces (see Theorem 11.4) are established for generalised weakly differentiable functions. The primary ingredient is
the Sobolev Poincaré inequality Theorem 10.1 (1).
First, some basic properties of approximate differentiation with respect to a
varifold are collected.
Lemma 11.1. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn , V ∈
RV m (U), Y is a finite-dimensional normed vector space, f is a kV k measurable Y valued function, and A is the set of points at which f is (kV k, m) approximately
differentiable.
Then, the following four statements hold:
(1) The set A is kV k measurable and (kV k, m) ap D f (x) ◦ Tanm (kV k, x)♮
depends kV k x A measurably on x .
(2) There exists a sequence of functions fi : U → Y of class 1 such that
kV k(A ∼{x : f (x) = fi (x) for some i}) = 0.
(3) If g : U → Y is locally Lipschitzian, then
kV k(U ∩ {x : f (x) = g(x)} ∼ A) = 0.
(4) If g is a kV k measurable Y -valued function and B = U ∩ {x : f (x) =
g(x)}, then B ∩ A is kV k almost equal to B ∩ dmn(kV k, m) ap D g , and
(kV k, m) ap D f (x) = (kV k, m) ap D g(x)
for kV k almost all x ∈ B ∩ A.
❐
Proof. Assume Y = R. Then, (1) is [Men12a, 4.5 (1)].
To prove (2), we may reduce the problem. First, we reduce it to the case
where kV k(U ∼ M) = 0 for some m-dimensional submanifold M of Rn of class 1
by [Fed69, 2.10.19 (4), 3.2.29]. Second, we reduce it to the case where, for some
1 < λ < ∞, the varifold satisfies additionally that λ−1 ≤ Θ m (kV k, x) ≤ λ for
kV k almost all x by [Fed69, 2.10.19 (4)] and Allard [All72, 3.5 (1a)]; third,
it is hence reduced to the case where Θ m (kV k, x) = 1 for kV k almost all x
by [Fed69, 2.10.19 (1) (3)]. Finally, we reduce it to the case kV k = L m by
[Fed69, 3.1.19 (4), 3.2.3, 2.8.18, 2.9.11] which may be treated by means of
[Fed69, 3.1.16].
Finally, (4) follows from [Fed69, 2.10.19 (4)], and implies (3) by [Men12a,
4.5 (2)].
Next, the existence of an approximate differential for generalised weakly differentiable functions is established, and the approximate differential is linked to
the generalised weak differential.
U LRICH M ENNE
1064
Theorem 11.2. Suppose m, n, U , and V are as in Hypothesis 6.1, Y is a finitedimensional normed vector space, and f ∈ T(V , Y ).
Then, f is (kV k, m) approximately differentiable with
m
V D f (a) = (kV k, m) ap D f (a) ◦ Tan (kV k, a)♮
at kV k almost all a.
Proof. In view of Lemma 11.1 (4), use Lemmas 8.12 and 8.17 to reduce the
n
problem to the case where f is bounded and V D f ∈ Lloc
1 (kV k, Hom(R , Y )).
In particular, one may assume Y = R by Lemma 8.18. Define β = ∞ if m = 1
and β = m/(m − 1) if m > 1. Let ga : dmn f → R be defined by ga (x) =
f (x) − f (a) for a, x ∈ dmn f .
First, the following assertion will be shown:
lim sup s −1 kV k(B(a, s))−1
s→0+
Z
B(a,s)
ga dkV k < ∞
for kV k almost all a.
For this purpose, define C = {(a, B(a, r )) : B(a, r ) ⊂ U} and consider a point a
satisfying for some M the conditions
lim sup s
−m
Z
sup{4, n} ≤ M < ∞,
1 ≤ Θ m (kV k, a) ≤ M,
(kV k + kδV k)(∞) (f ) ≤ M,
B(a,s)
s→0+
|V D f | dkV k + kδV k B(a, s) < M,
Θ m (kV k, ·) and f are (kV k, C) approximately continuous at a,
which are met by kV k almost all a by [Fed69, 2.10.19 (1) (3), 2.8.18, 2.9.13].
Define ∆ = Γ10.1 (M). Choose 1 ≤ Q ≤ M and 1 < λ ≤ 2 subject to the requirements Θ m (kV k, a) < 2λ−m (Q − 41 ) and
either
or
Q = Θ m (kV k, a) = 1
Q < Θ m (kV k, a).
Then, pick 0 < r < ∞ such that B(a, λr ) ⊂ U and
1
α(m)s m ,
2
kV k U(a, λs) ≤ 2(Q − M −1 )α(m)s m ,
kV k B(a, s) ≥
kV k(U(a, λs) ∩ {x : Θ m (kV k, x) < Q}) ≤ ∆−1 s m ,
Z
|V D f | dkV k + kδV k B(a, λs) ≤ Mλm s m
B(a,λs)
Weakly Differentiable Functions on Varifolds
1065
for 0 < s ≤ r . Choose y(s) ∈ R such that
1
kV k U(a, λs),
2
1
kV k(U(a, λs) ∩ {x : f (x) > y(s)}) ≤ kV k U(a, λs)
2
kV k(U(a, λs) ∩ {x : f (x) < y(s)}) ≤
for 0 < s ≤ r ; in particular,
|y(s)| ≤ M
and f (a) = lim y(s).
s→0+
Define fs (x) = f (x) − y(s) whenever 0 < s ≤ r and x ∈ dmn f . Recalling
Lemma 8.12, Example 8.13 (4), and Remark 9.2, one applies Theorem 10.1 (1a)
with U , G, f , and r replaced by U(a, λs), 0, fs+ , respectively fs− , and s to infer
(kV k x B(a, (λ − 1)s))(β) (fs )
Z
Z
|V D fs | dkV k +
≤∆
B(a,λs)
B(a,λs)
|fs | dkδV k ≤ κs m
for 0 < s ≤ r , where κ = 2∆M 2 λm . Noting that
1/β
y(s) − y s · kV k B a, (λ − 1)s
2 2
(λ − 1)s
(fs/2 )
≤ kV k x B a,
2
(β)
+ (kV k x B(a, (λ − 1)s))(β) (fs ) ≤ 2κs m ,
y(s) − y s ≤ 2m+1 κ α(m)−1/β (λ − 1)1−m s,
2 one obtains for 0 < s ≤ r that
|y(s) − f (a)| ≤ 2m+2 κ α(m)−1/β (λ − 1)1−m s,
(kV k x B(a, (λ − 1)s))(β) (ga ) ≤ κ(1 + 2m+3 M)s m .
Combining the assertion of the preceding paragraph with [Men09, 3.7 (i)]
applied with α, q, and r replaced by 1, 1, and ∞, one obtains a sequence of
locally Lipschitzian functions fi : U → R such that
[
kV k U ∼ {Bi : i ∈ P } = 0,
where Bi = U ∩ {x : f (x) = fi (x)}.
Since f − fi ∈ T(V ) with
V D f (a) − V D fi (a) = V D (f − fi )(a) = 0
for kV k almost all a ∈ Bi
❐
by Theorem 8.20 (1) (3), the conclusion follows from Example 8.7 and from
Lemma 11.1 (4).
U LRICH M ENNE
1066
Lemma 11.3. If m, n, U , V , ψ, and p are as in Hypothesis 6.1, p = m, and
a ∈ U , then
Tanm (kV k, a) = Tan(spt kV k, a).
❐
Proof. If 0 < r < ∞, U(a, 2r ) ⊂ U , ψ(U(a, 2r ) ∼{a})1/m ≤ (2γ (m))−1 ,
and x ∈ U(a, r ) ∩ spt kV k, then kV k B(x, s) ≥ (2mγ (m))−m s m for 0 < s <
|x − a| by [Men09, 2.5].
Under suitable summability hypotheses, correspondingly sharper differentiability results in Lebesgue spaces follow.
Theorem 11.4. Suppose m, n, p, U , V are as in Hypothesis 6.1, 1 ≤ q ≤ ∞,
Y is a finite-dimensional normed vector space, f ∈ T(V , Y ),
n
V D f ∈ Lloc
q (kV k, Hom(R , Y )),
C = {(a, B(a, r )) : B(a, r ) ⊂ U},
and X is the set of points in spt kV k at which f is (kV k, C) approximately continuous.
Then, kV k(U ∼ X) = 0, and the following four statements hold:
(1) If m > 1, β = m/(m − 1), and f ∈ Lloc
1 (kδV k, Y ), then
lim r −m
r →0+
Z
B(a,r )
|f (x) − f (a) − hx − a, V D f (a)i|
|x − a|
β
dkV kx = 0
for kV k almost all a.
(2) If m = 1, then f |X is differentiable relative to X at a with
D(f |X)(a) = V D f (a)| Tanm (kV k, a) for kV k almost all a.
(3) If q < m = p and ι = mq/(m − q), then
lim r −m
r →0+
Z
B(a,r )
|f (x) − f (a) − hx − a, V D f (a)i|
|x − a|
ι
dkV kx = 0
for kV k almost all a.
(4) If p = m < q, then f |X is differentiable relative to X at a with
D(f |X)(a) = V D f (a)| Tanm (kV k, a) for kV k almost all a.
Proof. Clearly, kV k(U ∼ X) = 0 by [Fed69, 2.8.18, 2.9.13].
Assume p = q = 1 in case of (1) or (2). In case of (4), also assume p > 1 and
q < ∞. Define α = β in case of (1), α = ι in case of (3), and α = ∞ in case of (2)
or (4). Moreover, let ha : dmn f → Y be defined by
ha (x) = f (x) − f (a) − hx − a, V D f (a)i
whenever a ∈ dmn f ∩ dmn V D f and x ∈ dmn f .
Weakly Differentiable Functions on Varifolds
1067
The following assertion will be shown:
lim s −1(s −m kV k x B(a, s))(α) (ha ) = 0 for kV k almost all a.
s→0+
In the special case where f is of class 1, in view of Example 8.7, it is sufficient to
prove that
lim s −1 (s −m kV k x B(a, s))(α) (Norm (kV k, a)♮ (· − a)) = 0
s→0+
for kV k almost all a; and if ε > 0 and Tanm (kV k, a) ∈ G(n, m), then
Θ m (kV k x U ∩ {x : | Norm (kV k, a)♮ (x − a)| > ε|x − a|}, a) = 0
in case of (1) by [Fed69, 3.2.16] and
B(a, s) ∩ spt kV k ⊂ {x : | Norm (kV k, a)♮ (x − a)| ≤ ε|x − a|}
for some s > 0
in case of (2) or (3) or (4) by Lemma 11.3 and [Fed69, 3.1.21]. To treat the
general case, one obtains a sequence of functions fi : U → Y of class 1 such that
[
kV k U ∼ {Bi : i ∈ P } = 0,
where Bi = U ∩ {x : f (x) = fi (x)}
from Theorem 11.2 and Lemma 11.1 (2). Define gi = f − fi , and notice that
gi ∈ T(V , Y ) and
V D gi (a) = V D f (a) − V D fi (a)
for kV k almost all a
for i ∈ P by Theorem 8.20 (3). Define Radon measures µi over U by
µi (A) =
µi (A) =
Z∗
A
Z∗
A
|V D gi | dkV k +
|V D gi |q dkV k
Z∗
A
|gi | dkδV k
in case of (1),
in case of (2) or (3) or (4)
whenever A ⊂ U and i ∈ P , and notice that µi (Bi ) = 0 by Theorem 8.20 (1) (or
alternatively by Theorem 11.2 and Lemma 11.1 (4)). The assertion will be shown
to hold at a point a satisfying for some i ∈ P that
f (a) = fi (a),
V D f (a) = V D fi (a),
lim s −1 (s −m kV k x B(a, s))(α) (fi (·) − fi (a) − h· − a, V D fi (a)i) = 0,
s→0+
Θ m (kV k x U ∼ Bi , a) = 0,
ψ({a}) = 0,
Θ m (µi , a) = 0.
U LRICH M ENNE
1068
These conditions are met by kV k almost all a in view of the special case and
[Fed69, 2.10.19 (4)]. Choosing 0 < r < ∞ with B(a, 2r ) ⊂ U and
1
α(m)s m ,
2
ψ U(a, 2r ) ≤ Γ10.1 (2n)−1 in case of (2) or (3) or (4)
kV k(U(a, 2s) ∼ Bi ) ≤
for 0 < s ≤ r , one infers from Remark 9.2 and Theorem 10.1 (1) with U , M , G,
f , Q, and r replaced by U(a, 2s), 2n, 0, gi , 1, and s that
(kV k x B(a, s))(α) (gi ) ≤ ∆µi (B(a, 2s))1/q
in case of (1) or (2) or (3),
(kV k x B(a, s))(α) (gi ) ≤ ∆s 1−m/q µi (B(a, 2s))1/q
in case of (4)
for 0 < s ≤ r , where ∆ = Γ10.1 (2n) in case of (1) or (2), ∆ = (m − q)−1 Γ10.1 (2n)
in case of (3), and ∆ = Γ10.1 (2n)1/(1/m−1/q) α(m)1/m−1/q in case of (4). Consequently,
lim s −1 (s −m kV k x B(a, s))(α) (gi ) = 0,
s→0+
and the assertion follows.
From the assertion of the preceding paragraph, one obtains
lim (s −m kV k x B(a, s))(α) (| · −a|−1 ha (·)) = 0 for kV k almost all a;
s→0+
in fact, if B(a, r ) ⊂ U and κ = sup{s −1 (s −m kV k x B(a, s))(α) (ha ) : 0 < s ≤ r },
then one estimates
(s −m kV k x B(a, s))(α) (| · −a|−1 ha )
≤ s −m/α
∞
X
2i s −1 (kV k x(B(a, 21−i s) ∼ B(a, 2−i s)))(α) (ha )
i=1
≤ 2κ
∞
X
i=1
2(1−i)m/α =
2κ
1 − 2−m/α
in case of (1) or (3) and (s −m kV k x B(a, s))(α) (| · −a|−1 ha ) ≤ κ in case of (2) or
(4) for 0 < s ≤ r . This yields the conclusion in case of (1) or (3), and yields that
f |X is differentiable relative to X at a with
D(f |X)(a) = V D f (a)| Tan(X, a) for kV k almost all a
in case of (2) or (4) by [Fed69, 3.1.22]. To complete the proof, note that X is
dense in spt kV k; hence, if p = m, then
by [Fed69, 3.1.21] and Lemma 11.3.
❐
Tan(X, a) = Tan(spt kV k, a) = Tanm (kV k, a) for a ∈ U
Weakly Differentiable Functions on Varifolds
1069
Remark 11.5. The usage of ha in the last paragraph of the proof is adapted
from the proof of [Fed69, 4.5.9 (26) (II)].
12. C OAREA F ORMULA
Introduction. In this section, rectifiability properties of the distributional
boundary of almost all superlevel sets of real-valued generalised weakly differentiable functions are established (see Corollary 12.2). The result rests on the
approximate differentiability of such functions (see Theorem 11.2). To underline
this fact, it is derived as a corollary to a general result for approximately differentiable functions (see Theorem 12.1).
Combining the classical coarea formula for real-valued functions on submanifolds both of class 1 with the appropriate functional analytic perspective, one
obtains the following coarea formula for approximately differentiable functions
on rectifiable varifolds.
Theorem 12.1. Suppose m, n ∈ P , m ≤ n, U is an open subset of Rn ,
V ∈ RV m (U), f is a kV k measurable real-valued function that is (kV k, m) approximately differentiable at kV k almost all points, F is a kV k measurable Hom(Rn , R)valued function with
m
Z
F (x) = (kV k, m) ap D f (x) ◦ Tan (kV k, x)♮
K∩{x : |f (x)|≤s}
′
for kV k almost all x,
|F | dkV k < ∞ whenever K is a compact subset of U and 0 ≤ s < ∞,
and T ∈ D (U × R, Rn ) and S(y) : D (U, Rn ) → R satisfy
Z
hϕ(x, f (x)), F (x)i dkV kx for ϕ ∈ D (U × R, Rn ),
Z
hθ, F i dkV k ∈ R for θ ∈ D (U, Rn )
S(y)(θ) = lim ε−1
T (ϕ) =
ε→0+
{x : y<f (x)≤y+ε}
whenever y ∈ R; that is, y ∈ dmn S if and only if the limit exists and belongs to R
for θ ∈ D (U, Rn ).
Then the following two statements hold:
(1) If ϕ ∈ L1 (kT k, Rn) and g is an R̄-valued kT k-integrable function, then
Z
S(y)(ϕ(·, y)) dL 1 y,
Z
ZZ
g dkT k =
g(x, y) dkS(y)kx dL 1 y.
T (ϕ) =
(2) There exists an L 1 measurable function W with values in RV m−1 (U) endowed with the weak topology such that, for L 1 almost all y ,
Tanm−1 (kW (y)k, x) = Tanm (kV k, x) ∩ ker F (x) ∈ G(n, m − 1),
Θ m−1 (kW (y)k, x) = Θ m (kV k, x)
U LRICH M ENNE
1070
for kW (y)k almost all x and
S(y)(θ) =
Z
hθ, |F |−1 F i dkW (y)k
for θ ∈ D (U, Rn ).
Proof. First, notice that Lemma 3.3 with J = R implies that T is representable
by integration and
T (ϕ) =
kT k(g) =
Z
Z
hϕ(x, f (x)), F (x)i dkV kx,
g(x, f (x))|F (x)| dkV kx
whenever ϕ ∈ L1 (kT k, Rn) and g is an R̄-valued kT k-integrable function.
Next, the following assertion will be shown: there exists an L 1 measurable
function W with values in RV m−1 (U) such that, for L 1 almost all y ,
Tanm−1 (kW (y)k, x) = Tanm (kV k, x) ∩ ker F (x) ∈ G(n, m − 1),
Θ m−1 (kW (y)k, x) = Θ m (kV k, x)
for kW (y)k almost all x and
T (ϕ) =
Z
g dkT k =
ZZ
ZZ
hϕ(x, f (x)), |F (x)|−1 F (x)i dkW (y)kx dL 1 y,
g(x, y) dkW (y)kx dL 1 y,
whenever ϕ ∈ L1 (kT k, Rn) and g is an R̄-valued kT k-integrable function. For
this purpose, choose a disjoint sequence of Borel subsets Bi of U , sequences Mi
n
of m-dimensional submanifolds
S∞ of R of class 1, and functions fi : Mi → R of
class 1 satisfying kV k(U ∼ i=1 Bi ) = 0 and
Bi ⊂ M i ,
fi (x) = f (x),
D fi (x) = (kV k, m) ap D f (x) = F (x)| Tanm (kV k, x)
whenever i ∈ P and x ∈ Bi (see Lemma 11.1,
S [Fed69, 2.8.18, 2.9.11, 3.2.17,
3.2.29] and Allard [All72, 3.5 (2)]). Let B = ∞
i=1 Bi . If y ∈ R satisfies
H m−1 (B ∩ {x : f (x) = y and (kV k, m) ap D f (x) = 0}) = 0,
Z
B∩K∩{x : f (x)=y}
Θ m (kV k, x) dH
m−1
x<∞
whenever K is a compact subset of U , then define W (y) ∈ RV m−1 (U) by
W (y)(k) =
Z
B∩{x : f (x)=y}
m−1
·Θ
k(x, ker(kV k, m) ap D f (x))
(kV k, x) dH
m−1
x
Weakly Differentiable Functions on Varifolds
1071
for k ∈ K (U × G(n, m − 1)). Applying [Fed69, 3.2.22] with f replaced by fi
and summing over i, one infers that
Z
A∩{x : s<f (x)<t}
=
Zt Z
s
|F | dkV k
A∩B∩{x : f (x)=y}
Θ m (kV k, x) dH
m−1
x dL 1 y
whenever A is kV k measurable and −∞ < s < t < ∞; hence,
Z
g(x, f (x))|F (x)| dkV kx =
ZZ
g(x, y) dkW (y)kx dL 1 y
whenever g is an R̄-valued kT k-integrable function. The remaining parts of the
assertion now follow by considering appropriate choices of g and recalling Example 2.23.
Consequently, (1) is implied by Theorem 3.4 (2) with J = R and Z = Rn .
Noting
S(y)(θ) = lim ε−1
ε→0+
Z y+ε Z
y
hθ, |F |−1 F i dkW (υ)k dL 1 υ
for θ ∈ D (U, Rn )
❐
whenever y ∈ dmn S , we see that (2) follows using Remark 2.2, Example 2.24,
and [Fed69, 2.8.17, 2.9.8].
The description of L 1 almost all distributional boundaries of the superlevel
sets occurring in the coarea formula (see Theorem 8.30) now readily follows.
Corollary 12.2. Suppose m, n, U , and V are as in Hypothesis 6.1, f ∈ T(V ),
and E(y) = {x : f (x) > y} for y ∈ R.
Then, there exists an L 1 measurable function W with values in RV m−1 (U)
endowed with the weak topology such that, for L 1 almost all y ,
Tanm−1 (kW (y)k, x) = Tanm (kV k, x) ∩ ker V D f (x) ∈ G(n, m − 1),
Θ m−1 (kW (y)k, x) = Θ m (kV k, x)
for kW (y)k almost all x and
V ∂E(y)(θ) =
Z
hθ, |V D f |−1 V D f i dkW (y)k
for θ ∈ D (U, Rn ).
❐
Proof. In view of Lemma 8.29 and Theorem 11.2, this is a consequence of
Theorem 12.1.
U LRICH M ENNE
1072
Remark 12.3. The formulation of Corollary 12.2 is modelled on similar results for sets of locally finite perimeter (see [Fed69, 4.5.6 (1), 4.5.9 (12)]). Observe, however, that the structural description of V ∂E(y) given here for L 1 almost all y does not extend to arbitrary kV k + kδV k measurable sets E such that
V ∂E is representable by integration; in fact, kV ∂Ek does not even need to be
the weight measure of some member of RV m−1 (U). (Using [Fed69, 2.10.28]
to construct V ∈ IV 1 (R2 ) such that kδV k is a nonzero Radon measure with
kδV k({x}) = 0 for x ∈ R2 and kV k(spt δV ) = 0, one may take E = spt δV .)
13. O SCILLATION E STIMATES
Introduction. In this section, two situations are studied where the oscillation
of a generalised weakly differentiable function may be controlled by its generalised
weak derivative, if we assume a suitable summability of the mean curvature of
the underlying varifold. In general, such control is necessarily rather weak (see
Theorem 13.1 and Remark 13.2), but under special circumstances one may obtain
Hölder continuity (see Theorem 13.3). The main ingredients are the analysis of
the connectedness structure of a varifold (see Corollary 6.14) and the Sobolev
Poincaré inequalities with several medians (see Theorem 10.9).
The next theorem estimates the oscillation of a continuous generalised weakly
differentiable functions under natural summability hypotheses on its derivative.
Clearly, such an estimate needs to depend strongly on the underlying varifold.
It will, however, still be sufficient to prove the finiteness result for the geodesic
distance in Theorem 14.2.
Theorem 13.1. Suppose m, n, p, U , and V are as in Hypothesis 6.1, p = m,
K is a compact subset of U , 0 < ε < ∞, U(x, ε) ⊂ Rn ∼ U for x ∈ K , and one of the
following hold:
(1) Either 1 = m = q and
Θ 1 (kV k, a) : a ∈ K};
λ = 2m+5 α(m)−1 Γ10.1 (2n) sup{Θ
(2) Or 1 < m < q and λ = ε.
Then, there exists a positive, finite number Γ with the following property: if Y is a
finite-dimensional normed vector space, f : spt kV k → Y is a continuous function,
f ∈ T(V , Y ), and κ = sup{(kV k x U(a, ε))(q) (V D f ) : a ∈ K}, then
|f (x) − f (χ)| ≤ λκ
whenever x, χ ∈ K ∩ spt kV k and |x − χ| ≤ Γ −1 .
Proof. Let ψ be as in Hypothesis 6.1. Abbreviate A = spt kV k and δ = 1 −
m/q, and also ∆1 = 2m+3 α(m)−1 and ∆2 = Γ10.1 (2n). Notice that Θ m
∗ (kV k, a) ≥
1
for
a
∈
A
by
Corollary
4.8
(4)
and
Remark
7.6.
If
m
=
1,
then
2
dmn Θ 1 (kV k, ·) = U,
Θ 1 (kV k, a) : a ∈ K} < ∞
∆3 = sup{Θ
Weakly Differentiable Functions on Varifolds
1073
by Corollary 4.8 (1) (2). If m > 1, then Θ m−δ (kV k, a) = 0 for a ∈ A by Corollary 4.9. Moreover, if a ∈ A, 0 < r < ∞, and U(a, r ) ⊂ U , then, by Corollary 6.14 (3), there exists 0 < s ≤ r such that A ∩ U(a, s) is a subset of the
connected component of A ∩ U(a, r /2) that contains a. Since K ∩ A is compact,
one may therefore construct j ∈ P and ai , si , and ri for i = 1, . . . , j such that
Sj
K ∩ A ⊂ i=1 U(ai , si ) and
ai ∈ K ∩ A,
0 < si ≤ ri ≤ ε,
A ∩ U(ai , si ) ⊂ Ci ,
U(ai , ri ) ⊂ U,
m
,
1
ri
1
ψ(U(ai , ri ) ∼{ai }) ≤ ∆−
kV k U(ai , ri ) ≥ α(m)
2 ,
2
2
ri−1 kV k B(ai , ri ) ≤ 4∆3 if m = 1,
∆1 Γ10.9 (4) (2n)m/δ riδ−m kV k U(ai , ri ) ≤ ε
if m > 1
for i = 1, . . . , j , where Ci is the connected component of A ∩ U(ai , ri /2) that
contains ai . Since K ∩ A is compact, there exists a positive, finite number Γ with
the following property. If x, χ ∈ K ∩A and |x−χ| ≤ Γ −1 , then {x, χ} ⊂ U(ai , si )
for some i.
To verify that Γ has the asserted property, suppose that Y , f , κ , x , and χ are
related to Γ as in the body of the theorem.
Assume κ < ∞. Since |y| = sup{α(y) : α ∈ Hom(Y , R), kαk ≤ 1} for
y ∈ Y by [Fed69, 2.4.12], one may also assume Y = R by Lemma 8.18. Choose
i such that {x, χ} ⊂ Ci . Choose N ∈ P satisfying
ri
1
N α(m)
2
2
m
≤ kV k U(ai , ri ) ≤
ri
1
(N + 1)α(m)
2
2
m
.
Applying Theorem 10.9 (2) (4) with U , M , Q, r , and X replaced by U(ai , ri ),
2n, 1, ri /2, and {ai } yields a subset Υ of R such that card Υ ≤ N + 1 and
[
r1
f U ai ,
⊂ {B(y, κi ) : y ∈ Υ},
2
where κi = ∆2 κ if m = 1 and κi S
= Γ10.9 (4) (2n)m/δ riδ κ if m > 1. Defining I
to be the connected component of {B(y, κi ) : y ∈ Υ} that contains f (a), one
infers
≤ ∆1 κi ri−m kV k U(ai , ri ) ≤ λκ
since {f (x), f (χ)} ⊂ f [Ci ] ⊂ I and I is an interval.
❐
|f (x) − f (χ)| ≤ diam I = L 1 (I) ≤ 2(N + 1)κi
Remark 13.2. Considering varifolds corresponding to two parallel planes,
it is clear that Γ may not be chosen independently of V . Also, the continuity
hypothesis on f is essential, as may be seen by considering V associated with two
transversely intersecting lines.
U LRICH M ENNE
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In case the relevant density ratios of the varifold are bounded above away
from 2, a significantly more robust estimate of the Hölder seminorm of the generalised weakly differentiable function may be obtained without an a priori continuity hypothesis.
Theorem 13.3. Suppose 1 ≤ M < ∞. Then, there exists a positive, finite number Γ with the following property. If m, n, p, U , V , and ψ are as in Hypothesis 6.1,
p = m, n ≤ M , 0 < r < ∞, A = {x : U(x, r ) ⊂ U},
ψ U(a, r ) ≤ Γ −1 ,
kV k U(a, s) ≤ (2 − M −1 )α(m)s m
for 0 < s ≤ r
whenever we have a ∈ A ∩ spt kV k, Y is a finite-dimensional normed vector space,
f ∈ T(V , Y ), C = {(x, B(x, s)) : B(x, s) ⊂ U}, X is the set of a ∈ A ∩ spt kV k
such that f is (kV k, C) approximately continuous at a, m < q ≤ ∞, δ = 1 − m/q,
and κ = sup{(kV k x U(a, r ))(q) (V D f ) : a ∈ A ∩ spt kV k}, then
|f (x) − f (χ)| ≤ λ|x − χ|δ κ
whenever x, χ ∈ X and |x − χ| ≤
where λ = Γ if m = 1 and λ = Γ 1/δ if m > 1.
r
,
Γ
Proof. Define
∆1 = 1 −
1
1/M
,
4M − 1
∆2 = sup{1, Γ10.9 (4) (2M)}M ,
Γ = 4(1 − ∆1 )−1 sup{4Γ10.1 (2M), ∆2 },
and notice that Γ ≥ 4(1 − ∆1 )−1 .
To verify that Γ has the asserted property, suppose m, n, p, U , V , ψ, M , r ,
A, Y , f , C , X , q, δ, κ , x , χ , and λ are related to Γ as in the body of the theorem.
In view of Lemma 8.18 and Corollary 10.3 (2) (4), one may assume Y = R.
Define s = 2|x − χ| and t = (1 − ∆1 )−1 s , and notice that 0 < s < t ≤ r and
kV k U(x, t) ≤ (2 − M −1 )α(m)(1 − ∆1 )−m s m
−m m
≤ (2 − (2M)−1 )α(m)∆m
s
1 (1 − ∆1 )
= (2 − (2M)−1 )α(m)(t − s)m ,
ψ U(x, t) ≤ Γ −1 ≤ Γ10.1 (2M)−1 .
Therefore, applying Theorem 10.9 (2) (4) with U , M , N , Q, r , and X replaced by
U(x, t), 2M , 1, 1, t − s , and 0 yields a subset Υ of R with card Υ = 1 such that
(kV k x U(x, s))(∞) (fΥ ) is bounded by (using Hölder’s inequality)
Γ10.1 (2M)(kV k x U(x, t))(1) (V D f ) ≤ 4Γ10.1 (2M)t δ κ
if m = 1,
Weakly Differentiable Functions on Varifolds
1075
and by
1/δ
Γ10.9 (4) (2M)m/δ t δ (kV k x U(x, t))(q) (V D f ) ≤ ∆2 t δ κ
if m > 1.
❐
Noting |f (x) − f (χ)| ≤ 2(kV k x U(x, s))(∞) (fΥ ) and t = 2(1 − ∆1 )−1 |x − χ|,
the conclusion follows.
14. G EODESIC D ISTANCE
Introduction. Reconsidering the support of the weight measure of a varifold
whose mean curvature satisfies a suitable summability of the mean curvature, the
oscillation estimate Theorem 13.1 is used to establish in this section that its connected components agree with the components induced by its geodesic distance
(see Theorem 14.2).
First, an example related to the sharpness of the results of the present section
is provided.
Example 14.1. Whenever 1 ≤ p < m < n, U is an open subset of Rn ,
and X is an open subset of U , there exists V related to m, n, U , and p as in
Hypothesis 6.1 such that spt kV k equals the closure of X relative to U , as is readily
seen taking into account the behaviour of ψ and V under homotheties (compare
[Men09, 1.2]).
The main theorem of this section is the following.
Theorem 14.2. Suppose m, n, U , V , and p are as in Hypothesis 6.1, p = m,
C is a connected component of spt kV k, and a, x ∈ C . Then, there exist −∞ <
b ≤ y < ∞ and a Lipschitzian function g : {υ : b ≤ υ ≤ y} → spt kV k such that
g(b) = a and g(y) = x .
Proof. In view of Corollary 6.14 (4), one may assume C = spt kV k. Whenever
c ∈ C , denote by X(c) the set of χ ∈ C such that there exist −∞ < b ≤ y < ∞
and a Lipschitzian function g : {υ : b ≤ υ ≤ y} → C such that g(b) = c and
g(y) = χ . Observe that it is sufficient to prove that c belongs to the interior of
X(c) relative to C whenever c ∈ C .
For this purpose, suppose c ∈ C , choose 0 < r < ∞ with U(c, 2r ) ⊂ U ,
and let K = B(c, r ). If m = 1, define λ as in Theorem 13.1 (1), and notice that
λ < ∞ by Corollary 4.8 (1). Define q = 1 if m = 1 and q = 2m if m > 1.
Choose 0 < ε ≤ r such that
λ sup{kV k(U(χ, ε))1/q : χ ∈ K} ≤ r ,
where λ = ε in accordance with Theorem 13.1 (2) if m > 1, and define
s = inf{Γ13.1 (m, n, m, U, V , K, ε, q)−1 , r }.
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Next, define functions fδ : C → R̄ by letting fδ (χ) for χ ∈ C and δ > 0
equal the infimum of sums
j
X
|xi − xi−1 |
i=1
corresponding to all finite sequences x0 , x1 , . . . , xj in C with x0 = c , xj = χ and
|xi − xi−1 | ≤ δ for i = 1, . . . , j and j ∈ P . Notice that
fδ (χ) ≤ fδ (ζ) + |ζ − χ|
whenever ζ, χ ∈ C and |ζ − χ| ≤ δ.
Since C is connected and fδ (c) = 0, it follows that fδ is a locally Lipschitzian
real-valued function satisfying Lip(fδ |A) ≤ 1 whenever A ⊂ U and diam A ≤ δ;
in particular, fδ ∈ T(V ) and kV k(∞) (V D fδ ) ≤ 1 by Example 8.7. One infers
fδ (χ) ≤ r
whenever χ ∈ C ∩ B(c, s) and δ > 0
from Theorem 13.1. It follows that C ∩B(c, s) ⊂ X(c); in fact, if χ ∈ C ∩B(c, s),
one readily constructs gδ : {υ : 0 ≤ υ ≤ r } → Rn satisfying
gδ (0) = c,
gδ (r ) = χ,
dist(gδ (υ), C) ≤ δ
Lip gδ ≤ 1 + δ,
whenever 0 ≤ υ ≤ r ,
❐
whence we deduce (by [Fed69, 2.10.21]) the existence of g : {υ : 0 ≤ υ ≤ r } → C
satisfying g(0) = c , g(r ) = χ , and Lip g ≤ 1, since im gδ ⊂ B(c, (1 + δ)r ).
Remark 14.3. Note that the deduction of Theorem 14.2 from Theorem 13.1
is adapted from Cheeger [Che99, Section 17] who attributes the argument to
Semmes (see also David and Semmes [DS93]).
Remark 14.4. In view of Example 14.1, it is not hard to construct examples
showing that the hypothesis p = m in Theorem 13.1 may not be replaced by
p ≥ q for any 1 ≤ q < m if m < n. Yet, for indecomposable V , the study of
possible extensions of Theorem 14.2 as well as related questions seems to be most
natural under the hypothesis p = m − 1 (see Topping [Top08]).
15. C URVATURE VARIFOLDS
Introduction. In this section, Hutchinson’s concept of curvature varifold is
rephrased in terms of the concept of weakly differentiable functions proposed in
the present paper (see Theorem 15.6). To indicate possible benefits of this perspective, a result on the differentiability of the tangent plane map is included (see
Corollary 15.9).
First, to phrase the definition of curvature varifolds in an invariant way, some
preparations are needed.
Weakly Differentiable Functions on Varifolds
1077
Hypothesis 15.1. Suppose n ∈ P and Y = Hom(Rn , Rn ) ∩ {σ : σ = σ ∗ }.
Then, T : Hom(Rn , Y ) → Rn denotes the linear map that is given by the composition (see [Fed69, 1.1.1, 1.1.2, 1.1.4, 1.7.9])
⊂
Hom(Rn , Y ) → Hom(Rn , Hom(Rn , Rn ))
L⊗1Rn
--- R ⊗ Rn ≃ Rn ,
≃ Hom(Rn , R) ⊗ Hom(Rn , R) ⊗ Rn ----------------------→
where thePlinear map L is induced by the inner product on Hom(Rn , R); hence,
n
T (g) = i=1 hui , g(ui )i whenever g ∈ Hom(Rn , Y ), and u1 , . . . , un form an
orthonormal basis of Rn .
Lemma 15.2. Suppose n, Y , and T are as in Hypothesis 15.1, n ≥ m ∈ P ,
M is an m-dimensional submanifold of Rn of class 2, and τ : M → Y is defined by
τ(x) = Tan(M, x)♮ for x ∈ M . Then,
h(M, x) = T (D τ(x) ◦ τ(x))
whenever x ∈ M.
Proof. Differentiating the equation τ(x) ◦ τ(x) = τ(x) for x ∈ M , one
obtains
τ(x) ◦ hu, D τ(x)i ◦ τ(x) = 0
for u ∈ Tan(M, x),
T (D τ(x) ◦ τ(x)) ∈ Nor(M, x)
for x ∈ M ; and, denoting by u1 , . . . , un an orthonormal base of Rn , one computes
τ(x) • (D g(x) ◦ τ(x)) =
n
X
hτ(x)(ui ), D g(x)i • τ(x)(ui )
i=1
= −g(x) •
n
X
hτ(x)(ui ), D τ(x)i(ui ) = −g(x) • T (D τ(x) ◦ τ(x))
whenever g : M → Rn is of class 1 and g(x) ∈ Nor(M, x) for x ∈ M .
❐
i=1
Lemma 15.3. Suppose n and Y are as in Hypothesis 15.1 and n > m ∈ P .
Then, the vector space Y is generated by {P♮ : P ∈ G(n, m)}.
Proof. If u1 , . . . , um+1 are orthonormal vectors in Rn , um+1 ∈ L ∈ G(n, 1),
and
❐
Pi = span{uj : j ∈ {1, . . . , m + 1} and j ≠ i} ∈ G(n, m)
Pm+1
for i = 1, . . . , m + 1, then L♮ + (Pm+1 )♮ = (1/m) i=
1 (Pi )♮ . Hence, one may
assume m = 1, in which case [Fed69, 1.7.3] yields the assertion.
The definition of curvature varifolds can now be stated.
U LRICH M ENNE
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Definition 15.4. Suppose n, Y , T are as in Hypothesis 15.1, n ≥ m ∈ P ,
m
U is an open subset of Rn , V ∈ IV m (U), X = U ∩ {x : Tan (kV k, x) ∈
m
G(n, m)}, and τ : X → Y satisfies τ(x) = Tan (kV k, x)♮ whenever x ∈ X .
Then, the varifold V is called a curvature varifold if and only if there exists
n
F ∈ Lloc
1 (kV k, Hom(R , Y )) satisfying
Z
h(τ(x)(u), F (x)(u)), D ϕ(x, τ(x))i + ϕ(x, τ(x))T (F (x)) • u dkV kx = 0
whenever u ∈ Rn and ϕ ∈ D (U × Y , R).
Remark 15.5. Using approximation and the fact that τ is a bounded function, one may verify that the same definition results if the equation is required for
every ϕ : U × Y → R of class 1 such that
Clos((U × Y ) ∩ {(x, σ ) : ϕ(x, σ ) ≠ 0 for some σ ∈ Y })
is compact. Extending T accordingly, one may also replace Y by Hom(Rn , Rn )
n
n
n
in the definition; in fact, if F ∈ Lloc
1 (kV k, Hom(R , Hom(R , R ))) satisfies
the condition of the modified definition, then im F (x) ⊂ Y for kV k almost
all x , as may be seen by enlarging the class of ϕ as before, and considering
ϕ(x, σ ) = ζ(x)α(σ ) for x ∈ U , σ ∈ Hom(Rn , Rn ), ζ ∈ D (U, R), and
α : Hom(Rn , Rn ) → R is linear with Y ⊂ ker α (compare Hutchinson [Hut86,
5.2.4 (i)]).
Consequently, Definition 15.4 is equivalent to Hutchinson’s definition in
[Hut86, 5.2.1]. The present formulation is motivated by Lemma 15.3.
The concept of curvature varifold may be naturally linked to that of generalised weak derivative of the “tangent plane” function τ . The nontrivial part of
this characterisation lies in the sufficiency of the generalised weak differentiability
of τ , which mainly rests on the second half of the approximate differentiability
result for such functions (see Theorem 11.2) and the second-order rectifiability of
the underlying integral varifold (see [Men13, 4.8]).
Theorem 15.6. Suppose n, Y , and T are as in Hypothesis 15.1, n ≥ m ∈ P ,
m
U is an open subset of Rn , V ∈ IV m (U), X = U ∩ {x : Tan (kV k, x) ∈ G(n, m)},
m
and τ : X → Y satisfies τ(x) = Tan (kV k, x)♮ whenever x ∈ X .
Then, V is a curvature varifold if and only if kδV k is a Radon measure absolutely
continuous with respect to kV k and τ ∈ T(V , Y ). In this case,
h(V , x) = T (V D τ(x)) for kV k almost all x,
Z
(δV )x (ϕ(x, τ(x))u) = h(τ(x)(u), V D τ(x)(u)), D ϕ(x, τ(x))i dkV kx
whenever ϕ : U × Y → R is of class 1 such that
Clos((U × Y ) ∩ {(x, σ ) : ϕ(x, σ ) ≠ 0 for some σ ∈ Y })
is compact and u ∈ Rn .
Weakly Differentiable Functions on Varifolds
1079
Proof. Suppose V is a curvature varifold and F is as in Definition 15.4. If
ζ ∈ D (U, R), γ ∈ E (Y , R), and u ∈ Rn , one may take ϕ(x, σ ) = ζ(x)γ(σ ) in
Definition 15.4 and Remark 15.5 to obtain
Z
γ(τ(x))h(τ(x)(u), D ζ(x)i + ζ(x)hF (x)(u), D γ(τ(x))i dkV kx
Z
= − ζ(x)γ(τ(x))T (F (x)) • u dkV kx.
In particular, taking γ = 1, one infers that kδV k is a Radon measure absolutely
continuous with respect to kV k with T (F (x)) = h(V , x), and kV k for almost
all x . It follows that
(δV )((γ ◦ τ)ζ · u) = −
Z
ζ(x)γ(τ(x))T (F (x)) • u dkV kx
for ζ ∈ D (U, R), γ ∈ E (Y , R), and u ∈ Rn . Along with the first equation, this
implies τ ∈ T(V , Y ) with V D τ(x) = F (x) for kV k almost all x by Remark 8.4.
To prove the converse, suppose kδV k is a Radon measure absolutely continuous with respect to kV k and τ ∈ T(V , Y ). Since τ is a bounded function,
n
V D τ ∈ Lloc
1 (kV k, Hom(R , Y )) by Definition 8.3 (1). To prove the equation
for the generalised mean curvature vector of V , in view of Theorem 11.2 and
[Men13, 4.8], it is sufficient to prove that
h(M, x) = T ((kV k, m) ap D τ(x) ◦ τ(x))
for kV k almost all x ∈ U ∩ M
whenever M is an m-dimensional submanifold of Rn of class 2. The latter equation, however, is evident from Lemma 11.1 (4) and Lemma 15.2, since
Tan(M, x) = Tanm (kV k, x) for kV k almost all x ∈ U ∩ M
by [Fed69, 2.8.18, 2.9.11, 3.2.17] and Allard [All72, 3.5 (2)].
Next, define f : X → Rn × Y by f (x) = (x, τ(x)) for x ∈ dmn τ , and
notice that f ∈ T(V , Rn × Y ) with
V D f (x)(u) = (τ(x)(u), V D τ(x)(u))
for u ∈ Rn
for kV k almost all x by Theorem 8.20 (2) and Example 8.7. The proof may be
concluded by establishing the last equation of the postscript. Using approximation
and the fact that τ is a bounded function yields that it is sufficient to consider
ϕ ∈ D (U × Y , R). Choose ζ ∈ D (U, R) with
Clos(U ∩ {x : ϕ(x, σ ) ≠ 0 for some σ ∈ Y }) ⊂ Int{x : ζ(x) = 1},
❐
and denote by γ the extension of ϕ to Rn × Y by 0. Now, applying Remark 8.4
with Y replaced by Rn × Y yields the equation in question.
Remark 15.7. The first paragraph of the proof is essentially contained in
Hutchinson in [Hut86, 5.2.2, 5.2.3], and is included here for completeness.
U LRICH M ENNE
1080
Remark 15.8. If m = n, then Θ m (kV k, ·) is a locally constant, integervalued function whose domain is U ; in fact, since τ is constant, V is stationary by
Theorem 15.6, and the asserted structure follows from Allard [All72, 4.6 (3)].
Finally, combining the preceding theorem with the differentiability theory of
generalised weakly differentiable functions, one obtains the following corollary.
Corollary 15.9. Suppose that m, n ∈ P , 1 < m < n, β = m/(m − 1),
U is an open subset of Rn , V ∈ IV m (U) is a curvature varifold,
m
X = U ∩ {x : Tan (kV k, x) ∈ G(n, m)},
and τ : X → Y satisfies τ(x) = Tanm (kV k, x)♮ whenever x ∈ X . Then, kV k
almost all a satisfy
r →0+
Z
B(a,r )
|τ(x) − τ(a) − hx − a, V D τ(a)i|
|x − a|
β
dkV kx = 0.
Proof. This is an immediate consequence of Theorems 15.6 and 11.4 (1).
❐
lim r −m
Remark 15.10. Using Theorem 11.4 (2), one may formulate a corresponding
result for the case m = 1.
Remark 15.11. Notice that one may deduce decay results for height quantities from this result by using [Men10, 4.11 (1)].
n
Remark 15.12. If 1 < p < m and V D τ ∈ Lloc
p (kV k, Hom(R , Y )) (see
Hypothesis 15.1), one may investigate whether the conclusion still holds with β
replaced by mp/(m − p).
A PPENDIX . A LTERNATE S OURCES
The book of Federer [Fed69] provides an excellent comprehensive treatment of bynow-classical geometric measure theory. A similar remark holds for Allard’s paper
[All72] on varifolds. Yet, some readers may prefer to initially consult alternative
sources that allow faster access to certain particular parts of the theory.
The following lists provide a survey of the results employed from [Fed69] and
[All72] along with alternate sources for the less basic material if available (which
is mostly the case).
Grassmann algebra (see [Fed69, Section 1]). Mostly, basic material is used.
1.1.1–1.1.4 Tensor products
1.4.1
Alternating forms and covectors
1.7.1
Polarities, inner products, and norms
1.7.3
Principal axis theorem and related material
1.7.5
Inner products on the space of m-vectors
1.7.9
Inner product on the space of homomorphisms between certain inner product spaces
Weakly Differentiable Functions on Varifolds
1081
General measure theory—basics (see [Fed69, Sections 2.1–2.6]). This material is considered truly classical.
2.2.2–2.2.3 Approximation properties of Borel regular measures
2.2.6
Positive integers and space of sequences
2.2.17
Push forward of Radon measures
2.3.5–2.3.6 Lusin’s theorem and consequences
2.4.2
Integral and integrable and summable functions
2.4.10
Measures defined by integration
2.4.12
Seminorms, Hahn-Banach theorem, Lebesgue spaces
2.4.18
Push forward of measures and integration
2.5.8
Radon-Nikodym theorem
2.5.14
Riesz’s representation theorem with Radon measures
2.5.19
Weakly topologised space of Daniell integrals
2.6.2
Fubini’s theorem
General measure theory—covering and differentiation (see [Fed69, Sections 2.8–2.9]). Concerning differentiation theory, the present paper employs the
unifying concept of Vitali relation mainly for notational consistence with [Fed69].
In particular, knowledge of the special case of centred balls would be mostly sufficient. For differentiation theory based on centred balls, there are many alternate sources; here, Evans and Gariepy [EG15] and Ambrosio, Fusco, and Pallara
[AFP00] are used.
2.8.1
Terminology for covering theorems
2.8.16
Covering relations and ϕ Vitali relations
2.8.17
Vitali relations for measures satisfying a weak “doubling-type” condition
2.8.18
Vitali relations on directionally limited spaces
2.9.1
Terminology for differentiation of measures
2.9.5–2.9.8 Differentiation theory for Borel regular measures finite on bounded
sets (see [EG15, Theorems 1.29–1.31] for Euclidean space)
2.9.11
Characterising measurability of a set in terms of its density properties (see [EG15, Theorem 1.35] for necessity for Lebesgue measure)
2.9.12
Densities, approximate limits, and approximate continuity all with
respect to a measure and a Vitali relation
2.9.13
Characterising measurability of a function in terms of its approximate continuity (see [EG15, Theorem 1.37] for the necessity for
Lebesgue measure)
2.9.19
Differentiation of functions of locally bounded variation on the real
line, in particular, monotone functions (see [AFP00, 3.28, 3.29])
2.9.20
Characterisations of absolutely continuous functions on the real line
(see [AFP00, 3.31, 3.32])
1082
U LRICH M ENNE
General measure theory—Carathéodory’s construction (see [Fed69, Section 2.10]). From this section, only a few selected items are employed.
2.10.19 Properties of the upper densities of a measure (see [Sim83, 3.2, 3.5,
3.6])
2.10.21 The Arzélà Ascoli theorem for Lipschitzian functions and Hausdorff
distance
2.10.28 Cantor-type sets in the real line (see [AFP00, 1.67] for the standard
Cantor set)
2.10.43 Kirszbraun’s theorem (see [KP99, 5.2.2])
Rectifiability—differentials and tangents (see [Fed69, Section 3.1]). Apart
from the mostly widely documented results related to Whitney’s extension theorem, the present paper also makes of the general concepts of tangent cones for
sets and differentiation of functions for which no obvious alternate source seems
available. These concepts are indispensable for the formulation of the results in
Section 11.
3.1.11
Higher-order differentials of functions
3.1.12
Covering with balls of radii varying in a Lipschitzian way (see [Zie89,
1.3.4])
3.1.13
Partition of unity with general Whitney-type estimates (see [KP99,
5.3.1–5.3.10] for a similar construction)
3.1.16
Approximate Stepanoff-type characterisation of C 1 -rectifiable functions
in the sense of Anzellotti and Serapioni (see [AS94, 2.5])
3.1.19
Equivalent conditions for submanifolds
3.1.21
Tangent and normal cones to sets
3.1.22
Differentiation of functions relative to sets
Rectifiability—area and coarea for Lipschitzian maps (see [Fed69, Section 3.2]). Apart from widely used material such as the area and coarea formulae
which may be found for instance in Krantz and Parks [KP08], the present paper
also employs notions of approximate tangent vectors and approximate differentials for which no obvious alternate source seems available.13 These notions are
indispensable for the formulation of the results in Section 11.
3.2.3
Area formula in Euclidean space (see [KP08, 5.1.1, 5.1.13])
3.2.16
Approximate tangent and normal vectors and approximate differentiation with respect to a measure and a dimension
3.2.17
Approximate tangents for Hausdorff measure on a bi-Lipschitzian image of a compact set
13The reader should not confuse the concept of approximate tangent plane in the sense of [KP08,
5.4.4] or [Sim83, 11.8] with the concept of approximate tangent vectors in the sense of [Fed69, 3.2.16]
(see page 8 of the Introduction).
Weakly Differentiable Functions on Varifolds
3.2.22
3.2.29
1083
Coarea formula for maps between appropriately rectifiable sets (see
[KP08, 5.4.8, 5.4.9] for the case of a Euclidean target)
Characterisation of countably (H m , m) rectifiable sets in terms of covering by submanifolds of class 1 (see [KP08, 5.4.3])
Homological integration theory—differential forms and currents (see also
[Fed69, Section 4.1]). Much of the material on distributions and regularisation
is widely documented. Here, basic properties of rectifiable currents and flat chains
are employed in Section 5. In fact, flat chains are only used in Remark 5.10 where
their indicated usage could be replaced by means of the boundary rectifiability
theorem (see [Fed69, 4.2.16 (2)] or [Sim83, 30.3]). In particular, Simon [Sim83]
provides sufficient knowledge on currents for the present paper.
4.1.1
4.1.2
4.1.3
4.1.5
4.1.7
4.1.14
4.1.15
4.1.20
4.1.28
4.1.30
Spaces of smooth functions and distributions (see [KP08, Section 7.1])
Regularisation (smoothing) of functions and distributions
Density result for the space of smooth functions with compact support
defined on a product
Distributions representable by integration
Currents, constancy theorem, normal currents, push forward of currents (see [Sim83, 26.1–26.14, 26.20, 26.21, 26.27])
Push forward of flat chains (see [Sim83, 26.25] for normal currents)
Locality of the push forward for flat chains (see [Sim83, 26.24] for
normal currents)
Flat chains and integral geometric measure with exponent 1 (see also
[Sim83, 26.29, 26.30] for normal currents)
Characterisation of rectifiable currents
Push forward of rectifiable currents (see [Sim83, 27.2 (3)])
Homological integration theory—deformation and compactness (see also
[Fed69, Section 4.2]). From this section, only a basic slicing result is employed.
4.2.1
Slicing normal currents by Lipschitzian real-valued functions (see also
[Sim83, 28.4–28.10])
Homological integration theory—normal currents of dimension n in Rn
(see also [Fed69, Section 4.5]). In this section, real-valued functions of locally
bounded variation on Rn (in particular, sets of locally finite perimeter) are treated
as locally normal currents. An approach avoiding the use of currents is presented
by Evans and Gariepy in [EG15].
4.5.6
4.5.9 (12) (13)
The Gauss-Green theorem for sets of locally finite perimeter (see
[EG15, Sections 5.7–5.8])
Coarea formulae for real-valued functions of locally bounded
variation (see [EG15, Theorem 5.9])
1084
4.5.9 (26)
4.5.11
4.5.15
U LRICH M ENNE
Differentiability of approximate and integral type for real-valued
functions of locally bounded variation (see [EG15, Theorems
6.1 and 6.4])
Characterisation of sets of locally finite perimeter in terms of the
integral geometric measure of the measure-theoretic boundary
(see [EG15, Theorem 5.23])
Embedding into Lebesgue and Hölder spaces for weakly differentiable functions (see [EG15, Theorems 4.8 and 4.10])
Applications to the calculus of variations (see [Fed69, Section 5]). The
regularity results cited from this section are only used in Remark 5.10.
5.3.20
Interior regularity of one-dimensional integral currents minimising and
positive elliptic integrand
5.4.15
Regularity of absolutely area-minimising rectifiable currents in dimension not exceeding seven and one codimension (see also [Sim83, Section 37])
Varifolds (see Allard [All72]). The largest part of the material on varifolds
employed in the present paper is covered by Simon in [Sim83].
2.5
Basic notation on facts for submanifolds on Euclidean space (see also
[Sim83, Section 7])
3.1
Basic definitions and notation for varifolds (see [Sim83, pp. 227–229])
3.5
Basic facts on rectifiable and integral varifolds (see [Sim83, Section 11,
Section 15, Section 38])
4.2
First variation of a varifold (see [Sim83, 39.1–39.3])
4.3
Basic notation for varifolds whose first variation is representable by integration (see [Sim83, 39.3, 39.4])
4.5–4.7 Varifolds contained in submanifolds mostly of the same dimension (see
[Sim83, Section 41] for some parts)
4.8 (4)
Relation of varifolds to area-minimising currents (see [Sim83, p. 227]
for a related discussion)
4.10
Cutting a varifold by a smooth functions
5.1
Monotonicity identity and some basic consequences (see [Sim83, Section 17, Section 40])
5.4
Measure-theoretic upper semicontinuity of density (see [Sim83, 40.6])
5.5 (1)
Rectifiability theorem for varifolds (see [Sim83, 42.4])
7.1+7.3 Isoperimetric inequalities (see [Sim83, Section 18])
8.6
Upper semicontinuity of density (see [Sim83, 17.8])
Acknowledgements. The author would like to thank Dr. Theodora Bourni,
Dr. Leobardo Rosales, Dr. Melanie Rupflin, and in particular Dr. Sławomir Kolasiński for discussions and suggestions concerning this paper. Moreover, the author is grateful to the referee for the careful reading of the initial manuscript, and
Weakly Differentiable Functions on Varifolds
1085
to Mario Santilli for spotting an omission in Definition 2.4 in a preprint version
of this paper. Finally, the author would like to thank the staff of the Indiana University Mathematics Journal for their very thorough copy editing of this paper.
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Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Am Mühlenberg 1
D-14476 Golm, Germany
AND
University of Potsdam
Institute for Mathematics
OT Golm, Karl-Liebknecht-Straße 24–25
D-14476 Potsdam, Germany
E- MAIL: [email protected]; [email protected]
K EY WORDS AND PHRASES : Rectifiable varifold, (generalised) weakly differentiable function, distributional boundary, decomposition, relative isoperimetric inequality, Sobolev Poincaré inequality,
approximate differentiability, coarea formula, geodesic distance, curvature varifold.
2010 M ATHEMATICS S UBJECT C LASSIFICATION: 49Q15 (46E35).
Received: November 12, 2014; revised: May 5, 2015.