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Applicable Analysis
Vol. 84, No. 9, September 2005, 867–876
Cosine families generated by second-order
differential operators on W1,1(0, 1) with
generalized Wentzell boundary conditions
ANDRÁS BÁTKAI*y, KLAUS-JOCHEN ENGELz and MARKUS HAASEx
yELTE TTK, Department of Applied Analysis, Pf. 120, H-1518 Budapest, Hungary
zUniversità di L’Aquila, Dipartmento di Matematica, Sezione Ingeneria,
Località Monteluco, I-67040 Roio Poggio, Italy
xUniversität Ulm, Abteilung Angewandte Analysis, D-89069 Ulm, Germany
Communicated by R.P. Gilbert
(Received 19 July 2004; in final form 14 April 2005)
Using the abstract framework [Bátkai, A. and Engel, K.-J., 2004, Abstract wave equations
with generalized Wentzell boundary conditions. Journal of Differential Equations, 207, 1–20.]
we show that certain second-order differential operators with generalized Wentzell boundary
conditions generate cosine families and hence also analytic semigroups on W1,1(0,1). This
complements the main result [Favini, A., Ruiz Goldstein, G., Goldstein, J.A., Obrecht,
E. and Romanelli, S., 2003, General Wentzell boundary conditions and analytic semigroups
on W1, p ð0, 1Þ. Applicable Analysis, 82, 927–935.] on the generation of an analytic semigroup
by the second derivative with generalized Wentzell boundary conditions on W1, p(0, 1) for
1 < p < 1.
Keywords: Sine and cosine families; Second-order differential operators; Wentzell boundary
conditions; Phase spaces
1991 Mathematics Subject Classifications: 47D09; 34G10; 35L05
1. Introduction
In this article we continue our studies initiated in [1] on the generation of cosine families
by operators with generalized Wentzell boundary conditions. While in [1] the main
abstract result was applied to certain second-order differential operators on C[0,1], in
the present work the underlying Banach space is W1,1(0, 1).
*Corresponding author. Email: [email protected]
Applicable Analysis ISSN 0003-6811 print: ISSN 1563-504X online ß 2005 Taylor & Francis Group Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/00036810500148952
868
A. Bátkai et al.
This article is organized as follows. In section 2 we briefly point out the connection
between second-order Cauchy problems and cosine families. Moreover, we recall our
framework and the main abstract result from [1]. In section 3 we then apply our
approach to certain second-order differential operators on W1,1(0, 1). In Appendix A
we collect some notions on sine and cosine families and prove a simple result on
groups of operators.
2. The abstract framework
Before introducing our abstract framework from [1] we recall that the second-order
Cauchy problem
(
ðACP2 Þ
u€ ðtÞ ¼ AuðtÞ,
uð0Þ ¼ u0 ,
t 0,
u_ ð0Þ ¼ u1
for a linear operator A on a Banach space X is ‘‘well-posed’’ if and only if A generates
a cosine family ðCðtÞÞt0 on X. In this case the unique solution of (ACP2) is given by
uðtÞ ¼ CðtÞu0 þ SðtÞu1
Rt
where SðtÞ :¼ 0 CðrÞdr denotes the associated sine family.
For further motivations as well as the basic properties and facts on cosine families,
sine families and the associated phase spaces we refer to [1], [2, Sections 3.14–16],
and [3, Section 2.8].
Here we are particularly interested in the case where A is a differential operator with
so-called generalized Wentzell boundary conditions on a function space. To investigate
these operators from an abstract and unified point of view we introduced in [1] the
following:
Abstract framework 2.1
(i)
(ii)
(iii)
(iv)
(v)
We consider
two Banach spaces X and @X, called ‘‘state space’’ and ‘‘boundary space’’, resp.;
a ‘‘boundary operator’’ L 2 LðX, @XÞ;
a closed, densely defined ‘‘maximal operator’’ Am : DðAm Þ X ! X;
a ‘‘feedback operator’’ B : DðBÞ X ! @X;
a ‘‘boundary dynamic operator’’ C 2 Lð@XÞ.
Using these spaces and operators we define on X the operator A Am with abstract
‘‘generalized Wentzell boundary conditions’’ by
Af :¼ Am f ,
DðAÞ :¼ f 2 DðAm Þ \ DðBÞ : LAm f ¼ Bf þ CLf :
Next we introduce the abstract ‘‘Dirichlet operator’’
L0 :¼ ðLjker Am Þ1: @X ! ker Am X
ð2:1Þ
Cosine families generated by second-order differential operators
869
which is (if it exists) characterized by
L0 x ¼ f
()
Am f ¼ 0,
L f ¼ x:
The main result from [1] is the following generation theorem for cosine families with
abstract Wentzell boundary conditions.
THEOREM 2.2
In the situation of the Abstract Framework 2.1, suppose that
(i) the restriction A0 :¼ Am jker L generates an exponentially Lipschitz continuous sine
family ðS0 ðtÞÞt0 on X (cf Definition A.1);
(ii) the feedback operator B : V0 ! @X is bounded, where V0 :¼ VðA0 Þ is defined by
(A.2);
(iii) the abstract Dirichlet operator L0 :¼ ðLjker Am Þ1 2 Lð@X, XÞ exists;
(iv) the abstract ‘‘Dirichlet–Neumann operator’’ N :¼ BL0 2 Lð@XÞ is bounded.
Then A defined by (2.1) generates a cosine family ðCðtÞÞt0 on X with phase space V X
(cf Definition A.2), where V :¼ V0 ker Am .
3. Second-order differential operators on W1,1(0, 1) with generalized Wentzell
boundary conditions
In order to further demonstrate the power of our abstract approach from [1], we treat
in this article second-order differential operators on W1,1(0, 1). Before doing so we
introduce some notation.
As usual L1(0, 1) denotes the space of all absolutely integrable functions on the
interval ½0, 1 equipped with the norm
Z
1
k f kL1 :¼
j f ðrÞj dr:
0
Moreover, Wk, 1 ½0, 1, k 2 N, denotes the Sobolev space of all functions in L1 ð0, 1Þ
having derivatives up to order k in L1 ð0, 1Þ, which we equip with the norm
Z
k f kWk, 1 :¼
1
j f ðrÞj þ j f
ðkÞ
ðrÞj dr:
0
Finally, it will be convenient to use in the sequel the notation
1
Wk,
0 ð0, 1Þ :¼
f 2 Wk, 1 ð0, 1Þ : f ð0Þ ¼ f ð1Þ ¼ 0 :
Note that for k ¼ 1, 2 this space can be equipped with the norm
Z
k f kWk, 1 :¼
0
0
1
f
ðkÞ
ðsÞ ds,
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A. Bátkai et al.
which is, by Poincaré’s inequality, equivalent to the canonical norm k kWk, 1 . Observe
that one can extend k kWk, 1 to a semi-norm on Wk, 1 ð0, 1Þ.
0
The main result of this article is the following:
THEOREM 3.1 Let 0 < p 2 W2;1 ð0, 1Þ, B 2 LðW2;1 ð0, 1Þ, C2 Þ, C 2 M2 ðCÞ and P 2
LðW2, 1 ð0, 1Þ, W1, 1 ð0, 1ÞÞ. Then the operator
Af :¼ pf 00 þ Pf ,
pð0Þ f 00 ð0Þ
f ð0Þ
3, 1
DðAÞ :¼ f 2 W ð0, 1Þ :
¼ Bf þ C
pð1Þ f 00 ð1Þ
f ð1Þ
ð3:1Þ
generates a cosine family on X :¼ W1;1 ð0, 1Þ with phase space W2, 1 ð0, 1Þ W1, 1 ð0, 1Þ.
To prove this result we will use Theorem 2.2. As a preparation we need the following
lemma. For the relevant definitions see Appendix A.
LEMMA 3.2
Let 0 < p 2 W2, 1 ð0, 1Þ. Then the operator
1
DðA0 Þ :¼ W3,
0 ð0, 1Þ
A0 f :¼ pf 00 ,
generates an exponentially Lipschitz continuous sine family on X ¼ W1, 1 ð0, 1Þ with
1
V0 :¼ VðA0 Þ ¼ W2,
0 ð0, 1Þ:
For the proof of this result we proceed in three steps.
Step 1
The operator
1
DðA00 Þ :¼ W3,
0 ð0, 1Þ
A00 f :¼ f 00 ,
ð3:2Þ
generates an exponentially Lipschitz continuous sine family on X ¼ W1, 1 ð0, 1Þ with
1
V00 :¼ VðA00 Þ ¼ W2,
0 ð0, 1Þ:
Proof
ð3:3Þ
By [4, Section 2] or [5, Section 2] the operator
A~ 00 f :¼ f 00 ,
DðA~ 00 Þ :¼ C20 ½0, 1 :¼
f 2 C2 ½0, 1 : f ð0Þ ¼ 0 ¼ f ð1Þ
generates a sine family ðS~ 00 ðtÞÞt0 on C½0, 1, which is given by
1
S~00 ðtÞ f ðsÞ ¼
2
Z
sþt
f~ðrÞdr,
t 0, s 2 ½0, 1:
ð3:4Þ
st
Here f~ denotes the odd, 2-periodic extension of f 2 C½0, 1, i.e.,
(
f~ðrÞ :¼
f ðrÞ,
r 2 ½0, 1Þ,
f ðrÞ,
r 2 ½1, 0Þ,
f~ ðr þ 2nÞ :¼ f~ðrÞ,
r 2 ½1, 1Þ, n 2 Z:
ð3:5Þ
871
Cosine families generated by second-order differential operators
Since ðS~00 ðtÞÞt0 leaves W1, 1 ð0, 1Þ C½0, 1 invariant we can restrict it to X obtaining in
this way an operator family ðS00 ðtÞÞt0 on X. We will now check that ðS00 ðtÞÞt0 is an
exponentially Lipschitz continuous sine family with generator A00 . To this end we
have to verify the conditions (i)–(iii) in Definition A.1.
1
Proof (i) Note that the range RgðS00 ðtÞÞ W1,
0 ð0, 1Þ, hence it suffices to show (A.1)
for (the operator semi-norm corresponding to) the semi-norm k kW1, 1 on W1, 1 ð0, 1Þ.
0
Let 0 s t and take f 2 X, then
1
S00 ðtÞ f S00 ðsÞ f ðÞ ¼
2
Z
s
f~ ðrÞ dr þ
Z
t
f~ ðrÞ dr :
þt
þs
Hence, by Corollary A.4 (i) and since f~ is odd, f~ 0 is even and both are 2-periodic
S00 ðtÞ f S00 ðsÞ f 1
W1,
0
1
2
Z
1
j f~ ðr sÞ f~ ðr tÞj dr þ
1
Z
Z
1
j f~ ðr þ tÞ f~ ðr þ sÞj dr
1
1
ðt sÞ 1
j f~ 0 ðrÞj dr ¼ 2 ðt sÞ k f kW1, 1 :
0
This proves that ðS00 ðtÞÞt0 is exponentially Lipschitz continuous with ! ¼ 0 and
M ¼ 2. (ii) and (iii) Observe that1 ½DðA~ 00 Þ ,! W1, 1 ð0, 1Þ ,! C½0, 1 and A00 ¼
A~ 00 jW1, 1 ð0, 1Þ . Hence [6, Propositions IV.1.15 and 2.17] imply that ðA~ 00 Þ ¼ ðA00 Þ and
Rð, A00 Þ ¼ Rð, A~ 00 ÞjW1, 1 ð0, 1Þ . Since A~ 00 generates a sine family this implies (ii). Moreover,
since by the above calculations ðS00 ðtÞÞt0 is strongly continuous and bounded on X,
Rð2 , A00 Þ f ¼
Z
1
et S~00 ðtÞ f dt ¼
0
Z
1
et S00 ðtÞ f dt
0
for all > 0 and f 2 W1, 1 ð0, 1Þ, where the first integral is understood in C[0, 1] and the
second one in W1, 1 ð0, 1Þ. This shows (iii).
1
Next we show (3.3). To this end we first verify that W2,
0 ð0, 1Þ V00 , i.e. that
h
i
Rþ 3 t ° A00 S00 ðtÞ f ¼ f~ 0 ð þ tÞ f~ 0 ð tÞ ½0, 1
1
2 W1,
0 ð0, 1Þ
ð3:6Þ
1
is well-defined and continuous for every f 2 W2,
0 ð0, 1Þ. To this end we first observe that,
3,
1
1
since f~ is odd and 2-periodic, S00 ðtÞ f 2 W0 ð0, 1Þ ¼ DðA00 Þ and A00 S00 ðtÞ f 2 W1,
0 ð0, 1Þ
for all t 0. Moreover,
kA00 S00 ðt0 Þ f A00 S00 ðtÞ f kW1, 1 k f~ 00 ð þ t0 Þ f~ 00 ð þ tÞkL1 þ k f~ 00 ð t0 Þ f~ 00 ð tÞkL1
0
kTðt0 Þf~ 00 TðtÞ f~ 00 kL1 þ kTðt0 Þf~ 00 TðtÞ f~ 00 kL1
! 0 as t ! t0 ,
1
Here ½DðAÞ :¼ ðDðAÞ, k kA Þ: Moreover, ‘‘X ,! Y’’ denotes the continuous imbedding of X in Y.
872
A. Bátkai et al.
since the left-shift group ðTðtÞÞt 2 R is strongly continuous on L1 ð1, 1Þ (cf the discussion
before Corollary A.4).
1
To show the converse inclusion suppose that f 2 W1,
0 ð0, 1Þ such that the function
1
0
~
defined in (3.6) is continuous. Then for g :¼ f 2 L ð1, 1Þ Corollary A.4 (ii) implies
1
g 2 W1, 1 ð1, 1Þ and therefore f ¼ f~ j½0, 1 2 W2,
0 ð0, 1Þ as claimed.
2, 1
Finally, for every ! 0 and f 2 W0 ð0, 1Þ we have
h
i
sup ke! r A00 S00 ðrÞ f kW1, 1 ¼ sup e! r k f~ 00 ð þ rÞ f~ 00 ð rÞkL1
r0
0
r0
h
i
sup e! r ðk f~ 00 ð þ rÞkL1 þ k f~ 00 ð rÞkL1 Þ
r0
¼ 4 k f 00 kL1 ¼ 4 k f kW2, 1 ,
0
where we used the fact that f~ 00 is odd and 2-periodic. Hence it follows that k kW2, 1 is
0
finer than k kV0 and hence by the open mapping theorem these two norms are
equivalent.
g
Step 2
Let 0 < a 2 W2, 1 ð0, 1Þ. Then the operator A~ 0 defined by
A~ 0 f :¼ aðaf 0 Þ0 ,
1
DðA~ 0 Þ :¼ W3,
0 ð0, 1Þ
generates an exponentially Lipschitz continuous sine family ðS~0 ðtÞÞt0 on X ¼
W1, 1 ð0, 1Þ with
1
V~ 0 :¼ VðA~ 0 Þ ¼ W2,
0 ð0, 1Þ:
Proof
.
.
We start noting that by [7, Corollaries VIII.9, 10 and Proposition IX.6]
u, v 2 W1, 1 ð0, 1Þ implies uv 2 W1, 1 ð0, 1Þ and ðuvÞ0 ¼ u0 v þ uv0 ,
2 C1 ðRÞ, u 2 W1, 1 ð0, 1Þ implies u 2 W1, 1 ð0, 1Þ and ð uÞ0 ¼ ð
0
uÞ u0 .
Using these facts we show that A~ 0 and A00 from Step 1 are similar. To this end we
define
Z
s
’ðsÞ :¼
0
1
dr,
aðrÞ
s 2 ½0, 1,
where without loss of generality we assume that ’ð1Þ ¼ 1. Then ’ : ½0, 1 ! ½0, 1 is
strictly increasing and surjective, hence invertible with ’, ’1 2 W3, 1 ð0, 1Þ. Next we
define the operator Q’ 2 LðXÞ by
Q’ f :¼ f ’:
Then it is easy to verify that Q’ is invertible with bounded inverse Q1
’ ¼ Q’1 .
Moreover, we have
3, 1
3, 1
DðQ’ A00 Q1
’ Þ ¼ Q’ DðA00 Þ ¼ Q’ W0 ð0, 1Þ ¼ W0 ð0, 1Þ,
Cosine families generated by second-order differential operators
873
where the last equality follows from the facts that ’ð jÞ ¼ ’1 ð jÞ ¼ j for j ¼ 0, 1, and
’, ’1 2 W3, 1 ð0, 1Þ. Moreover, ð’1 Þ0 ðsÞ ¼ a ’1 ðsÞ and hence
ð f ’1 Þ00 ¼ ½ða f 0 Þ ’1 0 ¼ ða2 f 00 þ aa0 f 0 Þ ’1
which implies that
1 00
2
00
0
0
0 0
~
Q’ A00 Q1
’ f ¼ ð f ’ Þ ’ ¼ a f þ aa f ¼ aðaf Þ ¼ A0 f :
ð3:7Þ
Thus, by similarity and Step 1, A~ 0 ¼ Q’ A00 Q1
generates a sine family ðS~0 ðtÞÞt0
’
1
~
given by S0 ðtÞ ¼ Q’ S00 ðtÞQ’ , which is easily verified to be exponentially Lipschitz
continuous. To finish the proof it now suffices to observe that
1
2, 1
V~ 0 ¼ Q’ V00 ¼ Q’ W2,
0 ð0, 1Þ ¼ W0 ð0, 1Þ,
which easily follows from the Definition A.2 by similarity.
g
pffiffiffi
2, 1
Step 3 (Proof of Lemma 3.2) Let a :¼ p. Then 0 < a 2 W ð0, 1Þ and by (3.7) we have
1
f 2 DðA~ 0 Þ ¼ DðA0 Þ ¼ W3,
0 ð0, 1Þ:
A~ 0 f ¼ A0 f þ aa0 f 0 ,
1
1, 1
Hence, A~ 0 and A0 differ only by a perturbation in LðW2,
ð0, 1ÞÞ ¼ LðV~ 0 , XÞ
0 ð0, 1Þ, W
and the assertion follows from Step 2 and [1, Theorem A.4.(i) and (ii)].
g
We are now in the position to give the
Proof of Theorem 3.1 We first assume that P ¼ 0. Then the operator A defined in
2
1, 1
(3.1) fits into
our
Abstract Framework 2.1 if we choose X :¼ W ð0, 1Þ, @X :¼ C ,
ð0Þ
and Lf :¼ ff ð1Þ
. Moreover, Am f :¼ pf 00 with domain DðAm Þ :¼ W3, 1 ð0, 1Þ is closed
and densely defined. Next we verify assumptions (i)–(iv) of Theorem 2.2.
First observe that the Dirichlet operator L0 : @X ! ker Am X exists and is given by
L0
x0
x1
:¼ x0 "0 þ x1 "1
where "0 ðsÞ :¼ 1 s, "1 ðsÞ :¼ s for s 2 ½0, 1. This shows (iii), while (iv) is satisfied since
RgðL0 Þ W2, 1 ð0, 1Þ ¼ DðBÞ. Next, (i) follows from Lemma 3.2, which also states that
1
1
2, 1
V0 :¼ VðA0 Þ ¼ W2,
ð0, 1Þ, C2 Þ LðW2,
0 ð0, 1Þ. Since, by assumption, B 2 LðW
0 ð0, 1Þ,
2
C Þ ¼ LðV0 , @XÞ this implies (ii).
Hence, we conclude by Theorem 2.2 that A generates a cosine family with phase
space V X, where V :¼ V0 ker Am . Since2 ker Am ¼ h"0 , "1 i with "k ð jÞ ¼ kj ,
1
2, 1
k, j ¼ 0, 1, we then obtain V ¼ W2,
ð0, 1Þ as claimed.
0 ð0, 1Þ h"0 , "1 i ¼ W
This proves the theorem for P ¼ 0. In particular, if P ¼ 0 then A generates an
exponentially Lipschitz continuous sine family with V :¼ VðAÞ ¼ W2, 1 ð0, 1Þ and
DðAÞ ¼ X. For P 2 LðW2, 1 ð0, 1Þ, W1, 1 ð0, 1ÞÞ ¼ LðV, XÞ the claim now follows from
[1, Theorem A.4.(iii)].
g
2
Here h i denotes the linear span.
874
A. Bátkai et al.
Theorem 3.1 leaves complete flexibility in the choices of B 2 LðW2, 1 ð0, 1Þ, C2 Þ,
C 2 M2 ðCÞ and P 2 LðW2, 1 ð0, 1Þ, W1, 1 ð0, 1ÞÞ. Next we give a more concrete version of it.
COROLLARY 3.3
Let 0 < p 2 W2, 1 ð0, 1Þ, q, r 2 W1, 1 ð0, 1Þ and j , j 2 C, j ¼ 0, 1. Then
Af :¼ pf 00 þ q f 0 þ rf ,
DðAÞ :¼ f 2 W3, 1 ð0, 1Þ : f 00 ð jÞ ¼ j f 0 ð jÞ þ j f ð jÞ, j ¼ 0, 1
generates a cosine family on W1, 1 ð0, 1Þ with phase space
In particular, the wave equation
8
utt ðt, sÞ ¼ pðsÞuss ðt, sÞ þ qðsÞus ðt, sÞ þ rðsÞ uðt, sÞ,
>
>
<
uss ðt, jÞ ¼ j us ðt, jÞ þ j uðt, jÞ,
ðWEÞ
>
>
:
uð0, sÞ ¼ u0 ðsÞ, ut ð0, sÞ ¼ u1 ðsÞ,
W2, 1 ð0, 1Þ W1, 1 ð0, 1Þ.
t 0, s 2 ½0, 1,
t 0, j ¼ 0, 1,
s 2 ½0, 1
with generalized Wentzell boundary conditions is well-posed on W1, 1 ð0, 1Þ.
Remark 3.4
(i) Since by [2, Theorem 3.14.17] the generator of a cosine family also generates
an analytic semigroup of angle =2, the operator A in Corollary 3.3, or,
more generally, in Theorem 3.1 generates an analytic semigroup on W1, 1 ð0, 1Þ.
By [6, section II.6] also this means that the first-order Cauchy problem
ðACP1 Þ
u_ ðtÞ ¼ AuðtÞ,
t 0,
uð0Þ ¼ u0
for A as above is ‘‘well-posed’’ on W1, 1 ð0, 1Þ.
(ii) In [8] it is shown by completely different methods that the second derivative
Af :¼ f 00 ,
DðAÞ :¼ f 2 W3, p ð0, 1Þ : f 00 ð jÞ ¼ j f 0 ð jÞ þ j f ð jÞ, j ¼ 0, 1
generates an analytic semigroup on W1, p ð0, 1Þ for all 1 < p < 1. Hence
Corollary 3.3 complements this result to the non-reflexive case p ¼ 1, allowing
also much more general second-order differential operators.
References
[1] Bátkai, A. and Engel, K.-J., 2004, Abstract wave equations with generalized Wentzell boundary
conditions. Journal of Differential Equations, 207, 1–20.
[2] Arendt, W., Batty, C.J.K., Hieber, M. and Neubrander, F., 2001, Vector-valued Laplace Transforms
and Cauchy Problems. Monographs Math., Vol. 96 (Basel: Birkhäuser Verlag).
[3] Goldstein, J.A., 1985, Semigroups of Operators and Applications (New York: Oxford University Press).
[4] Favini, A., Ruiz Goldstein, G., Goldstein, J.A. and Romanelli, S., 2001, The one dimensional wave
equation with generalized Wentzell boundary conditions, In: S. Aizicovici and N. Pavel (Eds),
Differential Equations and Control Theory. Lecture Notes in Pure and Appl. Math., Vol. 225 (New
York: Marcel Dekker), pp. 139–145.
Cosine families generated by second-order differential operators
875
[5] Xiao, T.-J. and Liang, J., 2003, A solution to an open problem for wave equations with generalized
Wentzell boundary conditions. Mathematische Annalen, 327, 351–363.
[6] Engel, K.-J. and Nagel, R., 2000, One-parameter Semigroups for Linear Evolution Equations. Graduate
Texts in Mathematics, Vol. 194 (New York: Springer-Verlag).
[7] Brezis, H., 1986, Analisi Funzionale (Naples: Liguori Editore).
[8] Favini, A., Ruiz Goldstein, G., Goldstein, J.A., Obrecht, E. and Romanelli, S., 2003, General Wentzell
boundary conditions and analytic semigroups on W1, p ð0, 1Þ. Applicable Analysis, 82, 927–935.
Appendix A
In this appendix we recall some notation concerning sine and cosine families and
present a simple lemma on strongly continuous groups of operators, which we need
in order to prove Theorem 3.1.
Definition A.1 We call an operator family ðS0 ðtÞÞt0 exponentially Lipschitz continuous
sine family with generator A0 if there exist M, ! 0 such that
(i)
S0 ðtÞ S0 ðsÞ M
Z
t
e!r dr for all 0 s t;
ðA:1Þ
s
(ii) 2 2 ðA0 Þ whenever
> !;
R1
(iii) Rð2 , A0 Þ ¼ 0 et S0 ðtÞ dt for all > !.
Note that each sine family ðS0 ðtÞÞt0 satisfies S0 ð0Þ ¼ 0. Hence it can be extended to
an odd, strongly continuous function ðS0 ðtÞÞt 2 R on R by setting S0 ðtÞ :¼ S0 ðtÞ, t 0.
Given a sine family ðS0 ðtÞÞt0 with generator A0 satisfying the estimate (A.1), we
choose some ! > ! and define the Banach space V0 :¼ VðA0 Þ by
V0 :¼ f 2 X0 : A0 S0 ðÞ f 2 C! ðRþ , X0 Þ ,
ðA:2Þ
k f k :¼ k f k þ sup ke! r A S ðrÞ f k :
V0
X
0 0
r0
X
kk
Here we used the notations X0 :¼ DðA0 Þ X and
C! ðRþ , X0 Þ :¼ F 2 CðRþ , X0 Þ : lim ke! r FðrÞkX ¼ 0 :
r!1
By Theorem 2.2 and [1, Theorem A.4.(iii)], V(A0 ) is closely related to the notion of
‘‘phase space’’ in the following sense.
Definition A.2 Let A generate a cosine family ðCðtÞÞt0 on the Banach space X. If V is a
Banach space satisfying ½DðAÞ ,! V ,! X such that
0 I
A :¼
,
DðAÞ :¼ DðAÞ V
A 0
generates a strongly continuous semigroup ðT ðtÞÞt0 on the space V :¼ V X, then V is
called a phase space associated with ðCðtÞÞt0 (or with A).
Note that by [2, Theorem 3.14.11] a phase space is unique.
We close this appendix by the following auxiliary results on operator groups.
876
A. Bátkai et al.
LEMMA A.3 Let G be the generator of a strongly continuous group ðTðtÞÞt 2 R on a
Banach space X satisfying kTðtÞk M for all t 2 R. Then the following holds
(i) kTðtÞg TðsÞgk M jt sj kGgk for all s, t 2 R and g 2 DðGÞ.
(ii) If the map R 3 t ° ½TðtÞg TðtÞg 2 ½DðGÞ is continuous, then g 2 DðGÞ.
Proof
(i) Assuming without loss of generality that t s this follows immediately from
Zt
kTðtÞg TðsÞgk kTðrÞGgk dr M ðt sÞ kGgk:
s
(ii) From the assumption it follows that
R 3 t ° HðtÞ :¼ TðtÞ TðtÞg TðtÞg ¼ Tð2tÞg g 2 ½DðGÞ
is continuous. Integrating H(t) from 0 to 1 in ½DðGÞ we obtain
Z1
Z1
HðtÞ ¼
Tð2tÞg dt g 2 DðGÞ:
0
0
Since by [6, Lemma II.1.3.(iii)] we have
R1
0
Tð2tÞg 2 DðGÞ the claim follows.
If we apply this result to the left-shift
½TðtÞ f ðrÞ :¼ f ðr þ tÞ on the Banach space
L1 ð1, 1Þ
:¼ f f
2 L1loc ðRÞ :
group
ðTðtÞÞt 2 R
Z
f is 2-periodicg,
k f kL1 :¼
defined
g
by
1
j f ðrÞjdr
1
we obtain the following result. Here we use the fact that ðTðtÞÞt 2 R is strongly
continuous, isometric and that its generator is given by3
Gf ¼ f 0 ,
DðGÞ ¼ L1 ð1, 1Þ \ W1, 1 ð1, 1Þ:
For g 2 L1 ð1, 1Þ the following assertions hold.
R1
R1
(i) If g 2 W1, 1 ð1, 1Þ, then 1 jgðr þ tÞ gðr þ sÞj dr jt sj 1 jg 0 ðrÞj dr.
(ii) If
the
map R 3 t ° gð þ tÞ gð tÞ 2 W1, 1 ð1, 1Þ
is
continuous,
1, 1
g 2 W ð1, 1Þ.
COROLLARY A.4
3
Here we identify a function in L1 ð1, 1Þ with its restriction to [1, 1].
then