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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 977 c , Vol. 65, No. 3 (2016) Indiana University Mathematics Journal 978 991 993 998 1002 1006 1010 1015 978 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 1021 1036 1048 1063 1069 1072 1075 1076 1080 1085 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]. Weakly Differentiable Functions on Varifolds 979 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 980 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. Weakly Differentiable Functions on Varifolds 981 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, U LRICH M ENNE 982 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 ) . Weakly Differentiable Functions on Varifolds 983 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 . U LRICH M ENNE 984 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♮ . Weakly Differentiable Functions on Varifolds 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. U LRICH M ENNE 986 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. 996 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 1000 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 1004 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 1006 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 , U LRICH M ENNE 1014 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), U LRICH M ENNE 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 1018 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. U LRICH M ENNE 1020 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. U LRICH M ENNE 1022 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. U LRICH M ENNE 1024 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 1026 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 1056 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 1058 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 1060 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 1062 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 1074 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 }. U LRICH M ENNE 1076 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 1078 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. R EFERENCES [ADS96] G ABRIELE A NZELLOTTI , S ILVANO D ELLADIO , AND G IUSEPPE S CIANNA , BV functions over rectifiable currents, Ann. Mat. Pura Appl. (4) 170 (1996), 257–296. http://dx.doi.org/10.1007/BF01758991 . MR1441622 (97m:49034). [AF03] ROBERT A. A DAMS AND J OHN J. F. F OURNIER, Sobolev Spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. 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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.