Elements of Topology and Functional Analysis

Home , Compact embedding, Functional analysis

Appendix A Elements of topology and functional analysis

Here, for the reader’s convenience, we collect fundamental concepts, definitions, and theorems used in this monograph, which are, however, rather standard and thus presented here mostly without proof, although some specific generalizations or modifications are accompanied by proofs. There are many textbooks and monographs on this subject, such as, e.g., [52, 84, 173, 175, 314, 344, 624].

A.1 Ordering

A binary relation, denoted by Ä,onasetX is called an ordering if it is reﬂexive (i.e., x Ä x for all x 2 X), transitive (i.e., x1 Ä x2 & x2 Ä x3 imply x1 Ä x3 for all x1; x2; x3 2 X) and antisymmetric (i.e., x1 Ä x2 & x2 Ä x1 imply x1 D x2). The ordering Ä is called linear if x1 Ä x2 or x2 Ä x1 always holds for every x1; x2 2 X. An ordered set X is called directed if for every x1; x2 2 X, there is x3 2 X such that both x1 Ä x3 and x2 Ä x3. Instead of x1 Ä x2, we also write x2 x1.Byx1 < x2 we understand that x1 Ä x2 but x1 6D x2. Having two ordered sets X1 and X2 and a mapping f W X1 ! X2, we say that f is nondecreasing (respectively nonincreasing) if x1 Ä x2 implies f .x1/ Ä f .x2/ (respectively f .x1/ f .x2/). We say that x1 2 X is the greatest element of the ordered set X if x2 Ä x1 for every x2 2 X. Similarly, x1 2 X is the least element of X if x1 Ä x2 for every x2 2 X. We say that x1 2 X is maximal in the ordered set X if there is no x2 2 X such that x1 < x2. Note that the greatest element, if it exists, is always maximal but not conversely. Similarly, x1 2 X is minimal in X if there is no x2 2 X such that x1 > x2. The ordering Ä on X induces the ordering on a subset A of X, given just by the restriction of the relation Ä. We say that x1 2 X is an upper bound of A X if x2 Ä x1 for every x2 2 A. Analogously, x1 2 X is called a lower bound of A if x1 Ä x2 for every x2 2 A. If every two elements x1; x2 2 X possess both a least upper bound and a greatest lower bound, denoted respectively by sup.x1; x2/ and

© Springer Science+Business Media New York 2015 579 A. Mielke, T. Roubícek,ˇ Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences 193, DOI 10.1007/978-1-4939-2706-7 580 A Elements of topology and functional analysis inf.x1; x2/ and called the supremum and the infimum of fx1; x2g, then the ordered set .X; Ä/ is called a lattice. Then the supremum and the infimum exist for every finite subset and are determined uniquely because the ordering is antisymmetric. The so-called Kuratowski–Zorn lemma [343, 631] says that if every linearly ordered subset of X has an upper bound in X, then X has at least one maximal 1 element. Having a directed set and another set X, we say that .x /2 is a net in . / X if there is a mapping ! X W 7! x . Having another net xQ Q Q 2Q in X,we say that this net is finer than the net .x /2 if there is a mapping j W Q ! such Q Q that for every 2 ,wehavexQ Q D xj./Q , and moreover, for every 2 , there is Q 2 Q large enough that j.Q 1/ whenever Q 1 Q . The set of all natural numbers N ordered by the standard ordering Ä is a directed set. The nets having N (directed by this standard ordering) as the index set are called sequences. Every subsequence of a given sequence can be simultaneously understood as a finer net.2

A.2 Topology

A topology T of a set X is a collection of subsets of X such that T contains the empty set and X itself, and with every finite collection of sets also their intersection, and also with every arbitrary collection of sets also their union. The elements of T are called open sets (or T-open, if we want to indicate explicitly the topology in question), while their complements are called closed.AsetX endowed with a topology T is called a topological space; sometimes we denote it by .X; T/ to refer to T explicitly. A collection T0 of subsets of X is called a base of a topology T if every T-open set is a union of elements of T0. Then T WD f [˛A˛ j8˛ W A˛ 2 T0 g is a topology (induced by the base T0). Having a subset A X, TjA WD f A \ B j B 2 T g is a topology on A, called a relative topology.Havingx 2 N X, we say that N is a neighborhood of x if there is an open set A such that x 2 A N;thesetofall neighborhoods of a point x is denoted by N .x/ or, more specifically, NT.x/ if the topology T needs to be specified. A topology T is called Hausdorff if 8x1; x2 2 X, x1 ¤ x2 9A1 2 N .x1/; A2 2 N .x2/: A1 \ A2 D;. We define the interior,theclosure, and the boundary of a set A respectively by n ˇ o ˇ int.A/ WD x 2 X ˇ 9N 2 N .x/ W N A ; (A.2.1a)

1This assertion is unfortunately highly nonconstructive unless X N, and is equivalent to the axiom ofS choice: for every set X and every collection fAxgx2X, ;¤Ax X, there is a mapping . / f W X ! x2X Ax such that f x 2 Ax for every x 2 X. 2 Indeed, having a sequence .xk/k2N and its subsequence .xk/k2N with some N N, one can put WD .N; Ä/, Q WD .N; Ä/,andj W Q ! the inclusion N N. A Elements of topology and functional analysis 581

n ˇ o ˇ cl.A/ WD x 2 X ˇ 8N 2 N .x/ W N \ A 6D; (A.2.1b)

@A WD cl.A/ n int.A/: (A.2.1c)

Having A B X, we say that A is dense in B if cl.A/ B. A topological space is called separable if it contains a countable subset that is dense in it. Having a net .x /2 in the topological space X, we say that it converges to a point x 2 X if for every neighborhood N of x, there is 0 2 large enough that x 2 N whenever 0; then we say also that x is the limit point of the net in question, and write lim2 x WD x or simply x ! x. This concept of convergence is called the Moore–Smith convergence [433]. Note that x 2 cl.A/ if and only if there is a net in A converging to x; in this case, we also say that x is attainable by a net from A. A point x 2 X is called a cluster point of the net .x /2 if for every neighborhood N of x and for every 0 2 , there is 0 such that x 2 N. Clearly, every limit point is a cluster point as well, but not conversely. Nevertheless, for every 3 . / cluster point x of a net x 2 , there exists a finer net fxQ Q gQ 2Q converging to x. Ordering of all topologies on a given set X is naturally by inclusion: having two topologies T1 and T2 onasetX, we say that T1 is finer than T2 or T2 is coarser than T1 if T1 T2 (or equivalently, if the identity on X is .T1; T2/-continuous). The adjectives “stronger” and “weaker” are sometimes used in place of “finer” and “coarser,” respectively. A function d W XX ! R1 is called a quasidistance on X if for all x1; x2; x3 2 X, d.x1; x2/ 0, d.x1; x2/ D 0 is equivalent to x1 D x2, and d.x1; x2/ Ä d.x1; x3/ C d.x3; x1/. A quasidistance that does not take values 1 is called a distance.Every distance d induces a topology T by a base ffx 2 X j d.x; x1/<"gjx1 2 X;">0g. A distance d W XX ! R is called a metric on X if d.x1; x2/ D d.x2; x1/ for all x1; x2 2 X. Conversely, a topology is called metrizable if there exists a metric that induces it. However, it should be emphasized that there exist nonmetrizable topologies. We say that a topological space is compact if every open cover admits a finite subcover. In terms of nets, the equivalent definition reads that every net has a cluster point. There are various useful modifications of this notion. We say that a topology is sequentially compact if every sequence in X admits a subsequence that converges in X. A metrizable topology is compact if and only if it is sequentially compact. A set A is called relatively (sequentially) compact if the closure of A is (sequentially) compact in X. A topological space is called locally (sequentially) compact if every point of it possesses a (sequentially) compact neighborhood.

3It sufﬁces to put Q WD N .x/ directed by the ordering Äand to take, for every Q D

.; / Q Q N 2 ,somexQ Q WD x 2 N with . 582 A Elements of topology and functional analysis

A mapping f W X ! Y is called continuous if f .x/ D lim2 f .x / whenever a 4 net .x /2 X converges to x 2 X. If this holds only for sequences, the mapping is called sequentially continuous. The image of a (sequentially) compact set via a (sequentially) continuous mapping is (sequentially) compact. In the special case that Y D R, (sequential) continuity of a functional X ! R refers to the canonical Euclidean topology induced by the metric jj.IfR is equipped with the topology TC Df.a; 1/ j a 2 R g (respectively T D f .1; a/ j a 2 R g), which is obviously coarser than the Euclidean topology, we say that the function f W X ! R is (sequentially) lower semicontinuous (respectively (sequentially) upper semicontinuous) with respect the topology on X in question. Obviously, f is upper semicontinuous if and only if f is lower semicontinuous. This is also intimately related to the canonical ordering of R and just means that f .x/ Ä lim inf2 f .x / whenever a net .x /2 X converges to x 2 X, where the limit inferior of a net .y /2 R (here used for y D f ) is defined as R sup2 inff y j g; if values in 1 are admitted, this supremum always exists, because is directed and R [f˙1gis a complete lattice.5 Analogously, one defines the limit superior of a net .y /2 R as inf2 supf y j g, . / . / and then f upper semicontinuous just means that f x lim sup2 f x whenever x ! x. If the topology on X is metrizable, all these definitions can equivalently use D N equipped with the canonical ordering. Theorem A.2.1 (Bolzano–Weierstrass). 6 A lower (respectively upper) semicontinuous function on a compact space attains its minimum (respectively maximum).

The projective topology generated on X by the collection of mappings f W X ! Y and topologies T on Y is the coarsest topology T on X making T 1.T / continuous all the mappings f W X ! Y .Also, D sup f ,Q where 1.T / . / T f DfA XI f A 2 g. The topology on the Cartesian product X is canonicallyQ understood as the projective topologyQ generated by the projections T ; Pr W X ! X ; this topology has a base f 2 A I8 2 W A 2 & A D X for all but a ﬁnite number of indices 2 g and is referred to as the Tikhonov product topology. The following important result relies on the Kuratowski–Zorn lemma if is uncountable:

4 N . . // N . / . / This is equivalent to saying that 8x 2 X, N 2 TY f x 9M 2 TX x : f M N.Also,it 1 is equivalent to 8A 2 TY : f .A/ 2 TX. Still alternatively, it is equivalent to that if x is a cluster point of a net .x /2 ,thenf .x/ is a cluster point of .f .x //2 . 5 . / . / Equivalently, f is lower semicontinuous if f x Ä supA2N .x/ inf f A for every x 2 X. 6B. Bolzano is usually credited with showing (rather intuitively) that a real continuous function of a bounded closed interval Œa; b R is bounded. In fact, the compactness of Œa; b was not rigorously known during Bolzano’s lifetime (1781–1848), since definitions of the real numbers with suitable completeness properties were invented only later. His results were forgotten, cf. also [76], and later rediscovered in a much more general and truly rigorous context by K. Weierstrass. A Elements of topology and functional analysis 583 Q 7 Theorem A.2.2 (Tikhonov [604]). The product X is compact if and only if all X are compact. A compactification of .U; T/ is a pair .U; i/ with U compact and i W U ! U a continuous mapping8 with i.U/ dense in U. Ordering of compactifications of U is defined as follows: for two compactifications .1U; i1/ and .2U; i2/ of U, we say that the former one is a finer compactification than the latter one (or equivalently, the latter one is coarser than the former one) and write .1U; i1/ .2U; i2/ (or briefly 1U 2U)ifthereisa continuous mapping W 1U ! 2U fixing U in the sense that ı i1 D i2. Atriple.K; Z; i/ is called a convex compactification of a topological space .U; T/ if Z is a Hausdorff locally convex space, K is a convex, compact subset of Z, i W U ! K is continuous, and i.U/ is dense in K.Ifi is also injective, .K; Z; i/ is called a Hausdorff convex compactification. We define the ordering of convex compactifications as follows: for two convex compactifications .K1; Z1; i1/ and .K2; Z2; i2/ of U, we say that .K1; Z1; i1/ is finer than .K2; Z2; i2/, and write .K1; Z1; i1/ 9 .K2; Z2; i2/, if there is an affine continuous mapping W K1 ! K2 fixing U. If the Z’s are ignored, the ordering of convex compactifications agrees with the usual ordering of compactifications. The concept of convex compactifications was first introduced in [518]; for a comprehensive treatment of convex-compactification theory, we refer to [520].

A.3 Locally convex spaces, Banach spaces, Banach algebras

On a (real) linear space V ,10 a nonnegative, symmetric, degree-1 homogeneous, R subadditive functional kkV W V ! is called a norm if it vanishes only at 0; often, 11 we write brieﬂy kkinstead of kkV if V is obvious from the context. The norm induces a metric .v1;v2/ 7!kv1 v2k, which further induces a topology, called

7 The German spelling “Tychonoff” is sometimes used even in the English literature according to the original reference [604]. 8In general topology, a narrower concept of compactification, requiring i to be a homeomorphic embedding, is generally adopted. For our purposes, it appears useful to accept a wider concept. 9 .1 1 / 1 . / 1 . / ; The adjective “affine” means 2 z C 2 Qz D 2 z C 2 Qz for every z Qz 2 K1, while “fixing U” means ı i1 D i2. 10 A real linear space means that V is endowed with a binary operation .v1;v2/ 7! v1 C v2 W V V ! V htat makes it a group, i.e., v1 C v2 D v2 C v1, v1 C .v2 C v3/ D .v1 C v2/ C v3, 9 0 2 V : v C 0 D v,and8v1 2 V 9v2: v1 C v2 D 0, and furthermore, it is equipped with a scalar multiplication .a; x/ 7! ax W RV ! V satisfying .a1 C a2/v D a1v C a2v, a.v1 C v2/ D av1 Cav2, .a1a2/v D a1.a2v/,and1v D v. The occasionally used notation x=a is self-explanatory and naturally defined as x=a WD .1=a/x. 11The above-mentioned properties of a norm mean respectively kvk0, kavkDjajkvk, kuCvkÄkukCkvk for every u;v 2 V and a 2 R,andkvkD0 ) v D 0. 584 A Elements of topology and functional analysis

< the strong topology on V . A subset A V is called bounded if supu2A kuk 1. A linear space equipped with a norm is called a normed linear space. 0 0 If the last property (i.e., kukV D ) u D ) is missing, we call such a functional a seminorm.HavingV equipped with a collection fjj˛g˛2S of seminorms jj˛ with an arbitrary index set S, we call V a locally convex space. Then ffu 2 V j8˛ 2 S0 Wju uQ j˛ Ä " gjS0 S finite;">0;uQ 2 V g is a base of a topology, and V is considered equipped with the corresponding topology. If juj˛ D 0 for all ˛ 2 S implies u D 0, then this topology is Hausdorff, and V is called a Hausdorff locally convex space. In particular, a normed linear space V has the Hausdorff topology generated by the base ffu 2 V jku uQ kÄ" gj"> 0; uQ 2 V g. A normed linear space is a Hausdorff locally convex space with only one seminorm, namely just its norm; this refers to the strong topology. A net .u /2 is then called a Cauchy net if for all ˛ 2 S and ">0, there is " 2 such that ju1 u2 j˛ Ä whenever 1 and 2 . If every Cauchy net converges, the locally convex space V is called complete. Complete normed linear n spaces are called Banach spaces [51]. An example of a Banach space isPR endowed . n 2/1=2 with the norm, denoted usually by jjinstead of kk, defined by jsjD iD1 si ; such a Banach space is called an n-dimensional Euclidean space. If V is a Banach space such that for every v 2 V , V ! R W u 7!ku C vk2 ku vk2 is linear, then V is called a Hilbert space. In this case, we define the inner product (also called scalar product)by 1 1 ujv Á ujv WD ku C vk2 ku vk2: (A.3.1) V 4 4

By assumption, .j/WV V !R is a bilinear form, which is obviously symmetric12 and satisﬁes .uju/ Dkuk2. For example, the Euclidean space Rn is a Hilbert space. Let us call a Banach space V uniformly convex if kukDkvkD1 uCv 8 ">09 ı>08 u;v2V W ) Ä 1 ı: (A.3.2) kuvk" 2

If also a multiplication between elements of a Banach space V is deﬁned that is (as a binary operation V V ! V ) continuous and makes the Banach space V also a commutative ring and that is associative, then V is called a commutative Banach algebra.13 Banach algebras containing a unity14 are called unital. An example of a Banach algebra is the Banach space BV.˝/ with pointwise multiplication. Moreover, there is a deep theory of algebras over complex numbers that has also important projections into commutative unital Banach algebras over the real

12This means that both u 7! .ujv/ and v 7! .ujv/ are linear functionals on V and .ujv/ D .vju/. 13This means that uv D vu, a.uv/ D .au/v D u.av/,andkuvkÄkukkvk for every u;v2V and a2R. 14This means that there exists 1 2 V such that 1v D v for every v 2 V . A Elements of topology and functional analysis 585 numbers considered in this book. In particular, a C-algebra V is deﬁned [216] as a Banach algebra over the complex numbers possessing also an involution operation,15 denoted by v 7! v. The subalgebra of self-adjoint elements WD f v 2 V j v D v g forms a Banach algebra. Every unital commutative (separable) C-algebra enjoys a so-called Gelfand representation as C.K/ with a (metrizable) compact K; here, in contrast to the rest of this book, C stands for the complex-valued continuous functions instead of real-valued.16

A.4 Functions and mappings on Banach spaces, dual spaces

Having two normed linear spaces V 1 and V 2 and a mapping A W V 1 ! V 2,wesay that A is continuous if it maps convergent sequences in V 1 to convergent ones in V 2, and is a linear operator if it satisfies A.a1v1 C a2v2/ D a1A.v1/ C a2A.v2/ for any a1; a2 2 R and v1;v2 2 V 1. Often we write briefly Av instead of A.v/.IfV 1 D V 2, a linear continuous operator A W V 1 ! V 2 is called a projector if A ı A D A.The set of all linear continuous operators V 1 ! V 2 is denoted by Lin.V 1; V 2/, being itself a normed linear space when equipped with the addition and multiplication by scalars defined respectively by .A1 C A2/v D A1v C A2v and .aA/v D a.Av/, and with the norm

v v = v : kAkLin.V 1;V 2/ WD sup kA kV 2 D supkA kV 2 k kV 1 (A.4.1) kvkV 1 Ä1 v¤0

Since R itself is a linear topological space,17 we can consider the linear space Lin.V ; R/, being also denoted by V and called the dual space to V . The original space V is then called predual to V . For an operator (now a functional) f 2 V , ;v v ; R we write h f i instead of f . The bilinear form h iV V W V V ! is called a ; ; canonical duality pairing. Instead of h iV V , we often write brieﬂy h i. Always, V is a Banach space if endowed with the norm (A.4.1), denoted often brieﬂy by ;v kk instead of kkV , i.e., k f k D supkvkÄ1h f i. Obviously, ˝ ˛ ˝ ˛ ˝ ˛ ; ; = ;v : f u Dkuk f u kuk Äkuk sup f Dkf k kuk (A.4.2) kvkÄ1

15The involution is to satisfy .v/ D v, .uCv/ D u C v, uv D vu, .av/ DNav,and kvvkDkvkkvk for every u;v 2 V and a 2 C with aN denoting the complex conjugate to a. 16More speciﬁcally, the compact K can be taken as the set of all nontrivial multiplicative functionals from equipped with the weak* topology of . Alternatively, K can be taken as the set of all maximal ideals of the algebra equipped with the appropriate topology; recall the standard deﬁnition that a linear subset I is called an ideal if v 2I and u 2 imply vu 2I. 17The conventional norm on R is the absolute value j j. 586 A Elements of topology and functional analysis

If V is a Hilbert space, then .u 7! .f ; u// 2 V for every f 2 V , and the mapping f 7! .u 7! .f ; u// identifies V with V . Then (A.4.2) turns into the so-called Cauchy–Schwarz inequality .f ; u/ Äkf kkuk.IfV Á V WD ŒV , the Banach space V is called reflexive. Every Hilbert space is reflexive. One can consider a normed linear space V equipped with the collection of v ;v seminorms f 7!jhf ijgf 2V , which makes it a locally convex space; its topology is referred to as the weak topology and is strictly coarser than the norm topology unless V is finite-dimensional. Instead of uk ! u weakly, we sometimes write uk * u. The (sequential) weakly lower (respectively upper) semicontinuous functionals are defined just as in Section A.2 with T D the weak topology. It should be emphasized that sequentially weakly lower semicontinuous functionals need not be weakly lower semicontinuous.18 19 Theorem A.4.1. If V is uniformly convex, uk ! u weakly, and kukk!kuk, then uk ! u strongly. Likewise, the Banach space V can be endowed with the collection of seminorms ff 7!jhf ;vijgv2V , which makes it a locally convex space, its topology being referred to as the weak* topology. It is always coarser than the weak topology. * Instead of fk ! f weakly*, we sometimes write fk*f . The duality pairing is continuous if V V is equipped with the weak*norm or normweak topology. and it is also separately (weak*,weak)-continuous.20 Proposition A.4.2. If V is separable, then so is the dual space V . Proposition A.4.3. Bounded sets in V are relatively weakly* compact. If V is separable, then they are also relatively sequentially weakly* compact. The last assertion, in particular, yields a broadly applicable theorem: Theorem A.4.4 (Banach selection principle [52]). In a Banach space with a separable predual, every bounded sequence contains a weakly* convergent subsequence.

Having two locally convex spaces V 1 and V 2 and an operator A 2 Lin.V 1; V 2/, we deﬁne the so-called adjoint operator A 2 Lin.V 2; V 1/ by the identity hA f ;viDhf ; Avi,tobevalidforeveryv 2 V 1 and f 2 V 2.IfV 1 and V 2

18 2 1.˝/ An illustrative counterexample (cf. [141, Remarks 2.3–4]) is k kL2.˝/ W V WD H ! R. If it were weakly lower semicontinuous, then it would have to be bounded from below on a neighborhood N of 0, but each such N contains a line, and this functional is not bounded below on any line. The sequential (lower semi)continuity is due to the compact embedding H1.˝/ L2.˝/ (Rellich’s theorem, Theorem B.4.2) and continuity of this functional in the L2.˝/ topology. 19See Fan and Glicksberg [182] for a thorough investigation and various modiﬁcations. 20 This means, written “sequentially,” that both limk!1 liml!1h fk; uliDhf ; ui and liml!1 limk!1h fk; uliDhf ; ui if fk ! f weakly* and uk ! u weakly. A Elements of topology and functional analysis 587 are normed linear spaces, then A 7! A realizes an isometric (i.e., norm-preserving) isomorphism between Lin.V 1; V 2/ and Lin.V 2; V 1/. The linear structure of V 1 and V 2 allows us to investigate the smoothness of A. We say that A W V 1 ! V 2 has a directional derivative at u 2 V in the direction A.uC"h/A.u/ h 2 V if there exists the limit lim"!0C " . If this limit depends linearly and continuously on the direction h, then we say that A has a Gâteaux differential [215]atu 2 V , denoted by DA.u/ 2 Lin.V 1; V 2/.IfA W R ! R, we will write simply A0 instead of DA. If the Gâteaux differential exists in every point, A is called Gâteaux differentiable and DA W V 7! Lin.V 1; V 2/. In the special case V 2 D R,a Gâteaux-differentiable functional ˚ W V 1 ! R has the differential D˚ W V 1 ! V 1. Moreover, if also

A.u C uQ / A.u/ ŒDA.u/uQ lim D 0; (A.4.3) kuQkV 1 !0 kuQ kV 1 then A is called Fréchet differentiable at the point u.

A.5 Basics from convex analysis

AsetK in a linear space is called convex if u C .1/v 2 K whenever u;v 2 K and 2 Œ0; 1, and it is called a cone (with the vertex at the origin 0) if v 2 K whenever v 2 K and 0. A functional f W V ! R is called convex if f .u C .1/v/ Ä f .u/ C .1/f .v/. A functional f W V ! R is convex (respectively lower semicontinuous) if and only if its epigraph epi.f / WD f.x; a/ 2 V RI a f .x/g is a convex (respectively closed) subset of V R. A lower semicontinuous functional f is convex if and only if Á 1 1 u1 C u2 f .u1/ f .u2/ f : 2 C 2 2 (A.5.1)

If u1 ¤ u2 implies (A.5.1) with strict inequality, then f is called strictly convex. A convex functional is proper if it is not identically 1 and never takes value 1. For a convex subset K of a locally convex space V and u 2 K, we deﬁne closed convex cones TK.u/ V and NK.u/ V , called respectively the tangent cone and the normal cone,by21 [ Á n ˇ o ˇ TK.u/ WD cl a.Ku/ and NK.u/ WD f 2V ˇ 8v 2TK.u/ Whf ;viÄ0 : a>0

21 Note that v 2 TK .u/ means precisely that u C akvk 2 K for suitable sequences fakgk2N R and fvkgk2N V such that limk!1 vk D v. 588 A Elements of topology and functional analysis

Again, the normal cone is always a closed convex cone in V .Adomain of a functional F W V ! R deﬁned as Dom F WD f v 2 V j F.v/ < 1g is convex or closed if F is convex or lower semicontinuous, respectively. The subdifferential of a convex functional F W V ! R is deﬁned as a convex closed subset of V : n ˇ o ˇ @F.v/ WD 2 V ˇ 8 vQ 2 V W F.vCv/Q F.v/ Ch; wQ i : (A.5.2)

For two convex functionals F1; F2 W V ! R, @.F1CF2/ @F1 C @F2 holds simply by the deﬁnition (A.5.2), while the full sum rule

@.F1CF2/ D @F1 C @F2 (A.5.3) holds if Dom F1 \ int.Dom F2/ ¤;or int.Dom F1/ \ Dom F2 ¤;; cf. also Example A.5.2. Let V be a Banach space and F W V ! R1 a proper, lower semicontinuous, and convex functional. Its Legendre–Fenchel transform F W V ! R1 is deﬁned via n ˇ o ˇ F./ WD sup h;viF.v/ ˇ v 2V : (A.5.4)

The Fenchel equivalences for subdifferentials read

2 @F.v/ , v 2 @F./ , F.v/ C F./ Dh;vi: (A.5.5)

By the deﬁnition (A.5.4)ofF, we always have the Fenchel–Young inequality

F.v/ C F./ h;vi for all v 2V and 2V : (A.5.6)

For a convex proper lower semicontinuous F,wehave

F WD .F/ D F: (A.5.7)

A Banach linear space V is called ordered by a relation if this relation is an ordering and, in addition, it is compatible with the linear and topological structure.22 It is easy to see that D WD f u 2 V j u 0 g is a closed convex cone that does not contain a line. Conversely, having a closed convex cone D V that does not contain a line, the relation deﬁned by u v, provided u v 2 D, makes V an ordered linear topological space. The so-called negative polar cone f f 2 V j8v 2 V Whf ;vi0 g deﬁnes an ordering on V if it does not contain a line; this ordering is then called the dual ordering.

22This just means the following four properties: u 0 and a 0 imply au 0, u 0 and v 0 imply uCv 0, u v implies uCw vCw for every w,andu 0 and u ! x implies x 0. A Elements of topology and functional analysis 589

Example A.5.1. Using the notation ıK for the so-called indicator function deﬁned as ( 0 if v 2 K; ıK.v/ WD (A.5.8) 1 otherwise; we illustrate the above deﬁnitions by the following formulas: @ı @ı Œ@ı 1 @ı @ı ; K D NK and K D TK and K D K D K D NK (A.5.9) @ı Œ 1 holding for every convex closed K. The last relation also implies K D NK ,so that in particular,

@ı .0/ Œ 1.0/ : K D NK D K (A.5.10)

Example A.5.2 (Inequality in the sum rule). The equality (A.5.3) indeed needs a qualiﬁcation of F1 and F2. In general, even the very extreme situation V D @.F1CF2/ .v/ § @F1.v/ C @F2.v/ D;C; is easily possible. It occurs, e.g., for v D 0 2 R D V for23 ( p v if v 0; F1.v/ D F2.v/ D 1 if v<0:

Example A.5.3 (Löwner ordering). The set of positive semideﬁnite .nn/-matrices forms a closed convex cone. The ordering of all .nn/-matrices by this cone is called the Löwner’s ordering. Example A.5.4 (Orderings on Banach algebras and their duals). The subset f u 2 V j9v 2V W u D vv g of a C-algebra V forms a closed convex cone not containing a line and thus induces an ordering, and the elements in this set are said to be nonnegative.24 This cone is also contained in a linear subspace of the so-called self-adjoint elements, i.e., in f u 2 V j u D u g, which is then ordered, too. The dual ordering then orders the Banach space of all self-adjoint bounded linear functionals on the C-algebra V .Hereaself-adjoint functional onaC-algebra means that it is real-valued on the self-adjoint elements of V .

23 Note that F1 C F2 D ı0, while the derivative of F1 tends to 1 for v ! 0C and the derivative of F2 tends to 1 for v ! 0, so that indeed, both @F1.0/ D;and @F2.0/ D;. And obviously, Dom F1 D Œ0; 1/ and Dom F2 D .1;0, so that both Dom F1 \ int.Dom F2/ D;and int.Dom F1/ \ Dom F2 D;, and the qualification for (A.5.3) we mentioned above indeed is not satisfied. 24This ordering can alternatively be defined by saying that v 0 if its spectrum .v/ WD f 2 C j non9.v111/1 g is nonnegative, i.e., belongs to Œ0; C1/. Appendix B Elements of Measure Theory and Function Spaces

Here we collect basic concepts and results from speciﬁc function spaces and measure theory used in this book. There are many textbooks and monographs on this subject, such as [4, 179, 339, 444].

B.1 General measures

For a set S and a -algebra1 S of its subsets, a -additive2 set function W S ! R [f˙1gis called a measure.Thevariation j j of is a function S ! R1 deﬁned by ˇ Xn ˇ j j.A/ WD sup j .Ai/j ˇ n2N; 81ÄiÄn;1ÄjÄn; i¤j W Ai2S; Ai\Aj D; : iD1

We say that has finite variation if the total variation j j.S/ is finite. The measures on .S; S/ that have finite total variation take values only from R and can naturally be added and multiplied by real numbers, which makes the set of all such measures a linear space. It can be normed by the variation jj.S/, which makes it a Banach space. We denote it by M.S; S/ (or simply M.S/ if S is to be self-understood). If S is also a topological space, a set function is called regular if 8A2˙ 8">0 9A1; A22˙:cl.A1/Aint.A2/ and j j.A2nA1/Ä". The subspace of regular measures from M.S/, called Radon measures, is denoted by M .S/;itisa Banach space if normed by the variation jj.S/. An example of a -algebra is the

1 SWe call S an algebra if ;2S, A 2 S ) S n A 2 S,andA1; A2 2 S ) A1 [ A2 2 S.Ifalso A 2 S for every mutually disjoint A 2 S, it is called a -algebra. i2N i S P i 2 . / . / S This means i2N Ai D i2N Ai for every mutually disjoint Ai 2 .

© Springer Science+Business Media New York 2015 591 A. Mielke, T. Roubícek,ˇ Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences 193, DOI 10.1007/978-1-4939-2706-7 592 B Elements of Measure Theory and Function Spaces

Borel -algebra,3 denoted by B.S/. In the case S D B.S/, elements of M.S; S/ are called Borel measures.IfS is compact, there isR an isometric isomorphism . / M . / ;v v . / v . / f 7! W C S ! S defined by setting h f iD S dx for all 2 C S ; 4 this is known as the Riesz theorem. Then kf kC.S/ Dj j.S/. C 1 1 C For a measure , we define its positive variation WD 2 j jC 2 .If D C , we say that is positive. We denote by M0 .S/ WD f 2 M .S/ j .S/ D 1; positive g the convex set of probability measures.Fors 2 S, the measure ıs 2 C M0 .S/ defined by hıs;viDv.s/ is called the Dirac measure supported at s 2 S. One way of generating a -algebra on S is by an outer measure, which is, by S definition, a -subadditive monotone function 2 ! R1 vanishing on the empty set.5 A subset E of S is called -measurable if .A/ D .A \ E/ C .A n E/ for every A S.The -measurable sets form a -algebra, and restricted to the measurable sets is a complete measure.6 If .S; S/ and .X; X/ are measurable spaces, i.e., S and X are -algebras on S and X, respectively, then a mapping f W S ! X is called measurable (more precisely .S; X/-measurable) if f 1.B/ WD f s 2 S j f .s/ 2 B g2S for all B 2 X. For mapping F W S ! 2X, where 2X denotes the power set, we write F W S Ã X to indicate that F is a set-valued map, i.e., for all s 2 S,wehaveF.s/ X.The set Gr F WD f .s; x/ j x 2 F.s/ g is referred to as a graph of F. A set-valued map F W S Ã X is called measurable (more precisely, .S; X/-measurable) if for all B 2 X, the preimage F1.B/ WD f s 2 S j F.s/ \ B ¤;glies in S. A mapping f W S ! X is a selection of F if for all s 2 S,wehavef .s/ 2 F.s/. For a set S,a-algebra S of its subsets, and a measure W S ! R [f˙1g,we . ; S; / S say that S is a -finiteS complete measure space if there exists fSkgk2N . /< S . / 0 with Sk 1 and S D k2N Sk such that S1 S2 2 and S2 D implies S1 2 S. Throughout the rest of this subsection, we assume .S; S; / to be qualified in this way. On the product space SX, we use the product -algebra S˝B.X/, which is the smallest -algebra containing all cylinders AB with A 2 S and B 2 B.X/. Theorem B.1.1. 7 Let .S; S; / be a -finite complete measure space and X a complete separable metric space8 and consider a set-valued map F W S Ã X. On X, we consider the Borel -algebra B.X/.

3This is defined as the smallest -algebra containing all open subsets of S. 4The integral via is defined as the limit of simple functions, similarly as in Section B.2. P 5 2S R . 1 / 1 . / This means that W ! 1 is to satisfy [jD1Aj Ä jD1 Aj for every collection .Aj/j2N,andA B ) .A/ Ä .B/,andalso .;/ D 0. 6This means that every subset of a null set is measurable: S Â N 2 ˙ and .N/ D 0 H) S 2 ˙. 7See, e.g., [30, Theorems 8.1.3 and 8.1.4]. 8Completeness in metric spaces means that every Cauchy sequence converges. In fact, here (and at some other places too) a specific metric is not important, and it suffices only to assume the existence of some metric that makes X a complete and separable space. Such spaces are called Polish. B Elements of Measure Theory and Function Spaces 593

(i) If F W S Ã X is measurable and for each s 2 S, the sets F.s/ are nonempty and closed, then there exists a measurable selection f W S ! XforF. (ii) A set-valued map F W S Ã X is measurable if and only if Gr F 2 S ˝ B.X/. Proposition B.1.2 (Filippov selection, generalized9). Let .S; S; / and X be as in Theorem B.1.1,letFW S Ã X be a measurable set-valued mapping with closed nonempty images, let G be the -algebra on Gr.F/ that is the restriction of S ˝ B.X/ to Gr.F/, i.e., G WD f C \ Gr.F/ j C 2 S ˝ B.X/ g, and let g W Gr.F/ ! R be .G; B.R//-measurable and satisfy

9 x 2 F.s/ W g.s; x/ D 0; 8 s 2 S W (B.1.1) g.s; / W F.s/ ! R is continuous:

Then there exists a measurable selection f W S ! X of F such that g.s; f .s// D 0 for all s 2 S. Proof. We define the set-valued mapping H W S Ã X via H.s/ Dfx 2 F.s/ j g.s; x/ D 0 g. By assumption, each H.s/ is nonempty. Moreover, since g.s; / is continuous, each H.s/ is closed. Thus, to apply Theorem B.1.1(ii), it suffices to show that H W S ! X is measurable. For this, we use Theorem B.1.1(i). Define the function ' W Gr.F/ ! RI .s; x/ !7 .s; g.s; x//. Then for all S 2 S and B 2 B.R/,wehave n ˇ o '1.SB/ D .s; x/ 2 Gr.F/ ˇ g.s; x/ 2 B \ .SR/ 2 S ˝ B.X/; since the first term lies in S ˝ B.X/ due to the measurability of g. Thus, ' W Gr.F/ ! R is measurable, and we conclude that Gr.H/ D '1.Sf0g/ 2 S ˝ B.X/, which is the desired measurability of H W S Ã X. ut

B.2 Lebesgue and Hausdorff measures

The d-dimensional Lebesgue outer measure L d./ on the Euclidean space Rd, d 1, is deﬁned as ˇ X1 Yd ˇ [1 L d. / k k ˇ Œ k ; k ::: Œ k ; k ; k k : A WD inf bi ai A a1 b1 ad bd ai Äbi kD1 iD1 kD1 (B.2.1)

9This is a variant of the slight extensions of Filippov’s theorem as given in [30, Theorems 8.2.9 and 8.2.10]. The difference is that our function g is deﬁned only on Gr F and not on SX. 594 B Elements of Measure Theory and Function Spaces

Measurable sets with respect to this outer measure are called Lebesgue measurable.10 Since L d./ is an outer measure, the collection ˙ of Lebesgue measurable 11 d subsets of ˝ forms a -algebra. The function L W ˙ ! R1 is called the Lebesgue measure. For a set ˝ 2 ˙, we say that a property holds almost everywhere on ˝ (abbreviated a.e. on ˝)on˝ if this property holds everywhere on ˝ with the possible exception of a set of Lebesgue-measure zero; referring to those x where this property holds, we also say that it holds at almost all x 2 ˝ (abbreviated a.a. x 2 ˝). A function u W Rd ! Rm is called (Lebesgue) measurable if u1.A/ WD fx 2 RdI u.x/ 2 Ag is Lebesgue measurable for every A 2 Rm open. d m m We call u W R ! R a simple if it takes only a finite numberR of values vi 2 R 1.v / . / v ˙ . / andPu i DfxI u x D ig2 ; then we define the integral Rd u x dx naturally L d. /v as finite Ai i. Furthermore, a measurable u is called integrable if there is a sequence of simpleR functions .uk/k2N such that limk!1 uk.x/ D u.x/Rfor a.a. x 2 ˝, . / R . / and limk!1 Rd uk x dx exists in . Then this limit is denoted by Rd u x dx, and we call it the (Lebesgue) integral of u. It is then independentR of the particular choice . / ˝ Rd Rof the sequence uk k2N.If is measurable, by ˝ u dx we mean naturally

Rd uQ dx, where uQ WD ˝ u.By A, we mean the characteristic function deﬁned by . / 1 . / 0 A x WD for x 2 A and A x WD for x 62 A. Let .X; /be a metric space. For a subset U X, let diam(U) denote its diameter, that is, diam(U/ WD supf .x; y/ j x; y 2 U g.Ford 0 and S X, we deﬁne

1 Á d=2 X .U / d H d. / diam i ; S WD lim Sinf1 (B.2.2) .1Cd=2/ ı!0 S U 2 iD1 i iD1 diam.Ui/<ı R 1 z1 t with denoting the gamma function .z/ WD 0 t e dt. Note that the inﬁmum in (B.2.2) depends monotonically on ı; hence the limit exists. It can be seen that H d./ is an outer measure. By general theory, its restriction to the corresponding -algebra of measurable sets is a measure. It is called the d-dimensional Hausdorff measure [261]. All Borel subsets of X are H d-measurable. If d is a positive integer and X D Rd, then the coefﬁcient d=2= .1Cd=2/ is the volume of the unit ball in Rd, and H d is just the d-dimensional Lebesgue measure L d.

10This just means that A Rd is Lebesgue measurable if L d.A/ D L d.A \ S/ C L d.A n S/ for every subset S Rd. For example, all closed sets are Lebesgue measurable, hence every open set too, as well as their countable unions or intersections, etc. 11In fact, ˙ is the so-called Lebesgue extension of the Borel -algebra, created by adding all subsets of sets having measure zero. It has, together with the function L d W ˙ ! R [f1g, ˙ (and is characterized by) the following four properties:Q (i) A open implies A 2 ; (ii) A D Œ ; Œ ; L d. / d . / L d a1 b1 S ad bd withP ai Ä bi implies A D iD1 bi ai ; (iii) is countably additive, L d L d. / i.e., k2N Ak D k2N Ak for every countable collection fAkgk2N of mutually disjoint d d sets Ai 2 ˙;(iv)A B 2 ˙ and L .B/ D 0 implies A 2 ˙ and L .A/ D 0. B Elements of Measure Theory and Function Spaces 595

B.3 Lebesgue spaces, Nemytski˘ı mappings

We now consider ˝ Rd measurable with L d.˝/ < 1. The notions of measurability and the integral of functions ˝ ! Rm can be understood as before, provided all these functions are extended on Rd n ˝ by 0. By Lp.˝I Rm/ we denote 12 m the set of all measurable functions u W ˝ ! R such that kukLp.˝IRm/ < 1, where ( qR p . / p 1 < ; u WD ˝ ju x j dx for Ä p 1 (B.3.1) Lp.˝IRm/ . / ess supx2˝ ju x j for p D1 with jjdenoting the Euclidean norm on Rm.Theessential supremum ess sup in (B.3.1) is deﬁned as

ess sup f WD inf sup f .x/: (B.3.2) . / 0 x2˝ measd N D x2˝nN

Replacing “sup” with “inf” in (B.3.2) yields the essential inﬁmum, denoted by ess inf. The set Lp.˝I Rm/ endowed with pointwise addition and scalar multiplication, p m is a linear space. Besides, kkLp.˝IRm/ is a norm on L .˝I R /, which makes it a Banach space, called a Lebesgue space. 1 R If a measure 2 M .˝/N possesses a density d 2 L .˝/, which means .A/ D . / ˝ A d x dx for every measurable A , then has a certain special property, namely it is absolutely continuous with respect to the Lebesgue measure13 and also the converse assertion is true: every absolutely continuous measure possesses a density belonging to L1.˝/. This fact is known as the Radon–Nikodým theorem [457, 496]. An important question is how to characterize concretely the dual spaces. The 2 “natural” dualityR pairing comes from the innerP product in L -spaces, which means ;v . / v. / v m v Rm hu iWD ˝ u x x dx, where u WD iD1 ui i is the inner product in . If 1

1 1 0 ab Ä ap C bp ; (B.3.3) p p0

12Here, however, we adopt the usual convention not to distinguish between functions that are equal a.e., so that strictly speaking, Lp.˝I Rm/ contains equivalence classes of such functions. 13 This means that 8">09ı>08A ˝ measurable: measd.A/ Ä ı H) j .A/jÄ". 596 B Elements of Measure Theory and Function Spaces one gets14 the Hölder inequality [272]15 Z ˇ ˇ ˇ . / v. /ˇ v 0 ; u x x dx Ä u Lp.˝IRm/ Lp .˝IRm/ (B.3.4) ˝ where p0 is the so-called conjugate exponent deﬁned as 8 < 1 for p D1; 0 =. 1/ 1< < ; p WD : p p for p 1 (B.3.5) 1 for p D 1: R In fact, the modiﬁcation of (B.3.4)forp D 1 or p D1is trivially ˝ ju vjdx Ä 1 kukL1.˝IRd/kukL1.˝IRd/. Iterating this in the case m D , we get the Hölder inequality for a product of K 2 scalar functions: Z ˇ ˇ ˇ YK ˇ YK XK 1 ˇ . /ˇ 1: uk x dx Ä uk Lpk .˝/ with D (B.3.6) ˝ p kD1 kD1 kD1 k

From (B.3.4), it can be shown that the dual space is isometrically isomorphic to 0 Lp .˝I Rm/ if 1 Ä p < 1. On the other hand, the dual space to L1.˝I Rm/ is substantially larger than L1.˝I Rm/.16 Applying the algebraic Young inequality (B.3.3) to the right-hand side of (B.3.6), we obtain another important inequality, the integral Young inequality Z ˇ ˇ ˇ YK ˇ XK 1 XK 1 ˇ . /ˇ 1: uk x dx Ä uk Lpk .˝/ with D (B.3.7) ˝ p p kD1 kD1 k kD1 k

An important geometric property of the norm (B.3.1)isthatfor1

0W lim L x2˝ ˇ uk.x/u.x/ " D 0: (B.3.8) k!1

R R 0 R 14 1 1 1 p p 1 p p0 Just simply kuk p kvk 0 ju vjdx Ä kuk p juj dxC 0 kvk 0 juj dx D 1. L .˝/ Lp .˝/ ˝ p L .˝/ ˝ p Lp .˝/ ˝ 15Originally, Hölder stated this in a less symmetric form for sums in place of integrals. 16The elements of L1.˝I Rm/ are indeed very abstract objects and can be identiﬁed with ﬁnitely additive measures vanishing on zero-measure sets; see Yosida and Hewitt [621]. B Elements of Measure Theory and Function Spaces 597

Naturally, convergence a.e. means that uk.x/ ! u.x/ for a.a. x2˝. Proposition B.3.1 (Various modes of convergence). (i) Every sequence converging a.e. converges also in measure. (ii) Every sequence converging in measure admits a subsequence converging a.e. (iii) Every sequence converging in L1.˝/ converges in measure. 1 Theorem B.3.2 (Lebesgue [353]). Let fukgk2N L .˝/ be a sequence converg- 1 1 inga.e.tosomeuandR juk.xR/jÄv.x/ for some v 2 L .˝/. Then u lives in L .˝/, . / . / ˝ and limk!1 A uk x dx D A u x dx for every A measurable. 1 Theorem B.3.3 (Fatou [183]). Let fukgk2RN L .˝/ be a sequence of non- 17 negative functions such that lim infk!1 ˝ uk.x/ dx < 1. Then the function x 7! lim infk!1 uk.x/ is integrable, and Z Z Á lim inf uk.x/ dx lim inf uk.x/ dx: (B.3.9) k!1 ˝ ˝ k!1

Useful generalizations of the Lebesgue dominated convergence theorem (The- orem B.3.2) and the Fatou theorem (Theorem B.3.3) use sequences of upper or lower bounds that are uniformly integrable: a set M L1.˝/ (or more generally, of L1.˝I Rm/)isuniformly integrable if Z 8">0 9K 2 RC W sup ju.x/j dx Ä ": (B.3.10) u2M fx2˝Iju.x/jKg

18 1 Theorem B.3.4 (Vitali [613] ). Let fukgk2N L .˝/ be a sequence converging p p p a.e. to some u. Then u 2 L .˝/ and uk ! uinL .˝/ if and only if fjukj gk2N is uniformly integrable. Theorem B.3.5 (Fatou, generalized19). The conclusion of Theorem B.3.3 holds if uk 0 is replaced by uk vk with fvkgk2N uniformly integrable. From Theorem A.4.4, it immediately follows that bounded sets in Lp.˝I Rm/ are weakly or weakly* relatively sequentially compact, provided 1

17Obviously, existence of a common integrable minorant can simplify (but weaken) this assertion. 18More precisely, in [613], the integration of summable series is investigated rather than mere sequences. 19See Ash [27, Thm.7.5.2]. 598 B Elements of Measure Theory and Function Spaces measurable functions. Let f W X ! R1 be theR functionR deﬁned by f .x/ D . / limn!1 fn x . Then f is measurable, and limn!1 X fn dx D X f dx. Theorem B.3.7 (Lusin [365]). Every measurable function Œa; b ! C is a continuous function on nearly all its domain.20 Theorem B.3.8 (Dunford and Pettis [166]). Let M L1.˝I Rm/ be bounded. Then the following statements are equivalent: (i) M is relatively weakly compact in L1.˝I Rm/; (ii) the set M is uniformly integrable. The fundamental phenomenon that composition of two measurable mappings needR not be measurable can be handled in the context of integral functionals of the m type ˝ a.x; u.x// dx by qualifying a W ˝R ! R1 as a normal integrand if m 21 a.x; / W R ! R1 is lower semicontinuous for a.a. x 2 ˝ and a is measurable. m m A special case of a normal integrand a W ˝R ! R1 with a.x; / W R ! R continuous for a.a. x 2 ˝ is called a Carathéodory integrand. Having guaranteed measurability of a.x; u.x//, we can ask about integrability. The scalar character of m m0 values no longer plays any role, so we can consider a.x; / W R ! R , m0 1. Now, however, the growth of ja.x; /j plays a role, and it is thus worth allowing m m1 m some “anisotropy” by splitting R D R R j with some integers j, m1, m1 mj m0 ..., mj. We say that a W ˝R R ! R is a Carathéodory mapping m0 m1 mj if a.; r1;:::;rj/ W ˝ ! R is measurable for all .r1;:::;rj/ 2 R R and a.x; / W Rm1 Rmj ! Rm0 is continuous for a.a. x 2 ˝. Then the so-called mi Nemytski˘ı mapping Na maps functions ui W ˝ ! R , i D 1;:::;j, to a function m0 Na.u1;:::;uj/ W ˝ ! R deﬁned by Na.u1;:::;uj/ .x/ D a x; u1.x/;:::;uj.x/ : (B.3.11)

It should be emphasized that the composition of two measurable mappings need not be measurable, and thus the continuity of a.x; / is an important assumption. Theorem B.3.9 (Nemytski˘ı mappings on Lebesgue spaces). Let a W ˝Rm1 mj m0 R ! R be a Carathéodory mapping and let the functions ui W ˝ ! mi R ,iD 1;:::;j, be measurable. Then Na.u1;:::;uj/ is measurable. Moreover, if a also satisﬁes the growth condition

Xj ˇ ˇ ˇ ˇ = ˇ ˇ ˇ ˇpi p0 p0 a.x; r1;:::;rj/ Ä .x/ C C ri for some 2L .˝/; (B.3.12) iD1

20More speciﬁcally, this means that for an interval Œa; b,letf W Œa; b ! C be a measurable function. Then given ">0, there exists a compact E Œa; b such that f restricted to E is continuous and .Œa; bnE/<". Note that E inherits the subspace topology from Œa; b; continuity of f restricted to E is deﬁned using this topology. 21More precisely, this means that for each A open in Rm, f x 2 ˝ j epif .x; /\A ¤;gis Lebesgue measurable, i.e., the set-valued mapping t 7! epif .x; / is Lebesgue measurable. B Elements of Measure Theory and Function Spaces 599 with 1 Ä pi < 1, 1 Ä p0 < 1, then Na is a bounded continuous mapping p1 m1 pj mj p0 m0 L .˝I R /L .˝I R / ! L .˝I R /.Ifsomepi D1,iD 1;:::;j, the = same holds if the terms jjpi p0 are replaced by any continuous function.

B.4 Sobolev spaces

We now consider ˝ Rd open with L d.˝/ < 1; such a set is called a domain. p.˝/ @k =@ k1 :::@ kd For a function u 2 L , we define its distributional derivative u x1 xd with k1CCkd D k and ki 0 for every i D 1;:::;n as a distribution such that D E Z @ku @kg 8g2D.˝/ W ; g D .1/k u dx; (B.4.1) k1 k k1 k @ :::@ d ˝ @ :::@ d x1 xd x1 xd where D.˝/ stands for infinitely differentiable functions with compact support.The d-tuple of the first-order distributional derivatives @ u;:::; @ u is denoted by ru @x1 @xd and called the gradient of u.Forp < 1, we define a Sobolev space n ˇ o ˇ W1;p.˝/ WD u 2 Lp.˝/ ˇ ru2Lp.˝I Rd/ ; equipped with the norm

(B.4.2a) 8q < p p p < u Lp.˝/ C ru Lp.˝IRd/ if p 1 u 1; WD (B.4.2b) W p.˝/ : 1 ; : max kukL .˝/ krukL1.˝IRd/ if p D1

Since by Rademacher’s theorem, Lipschitz functions are a.e. differentiable, we have W1;1.˝/ D C0;1.˝/. Analogously, for k >1, we deﬁne n ˇ o ˇ k Wk;p.˝/ WD u2Lp.˝I Rm/ ˇ rku2Lp.˝I Rd / ;; (B.4.3) where rku denotes the set of all kth-order partial derivatives of u understood in the ; p k p.˝/ ; . distributional sense. The standard norm on W is kukWk p.˝/ D kukLp.˝/ C krkukp /1=p, which makes it a Banach space. Likewise for Lebesgue spaces, Lp.˝IRdk / for 1 Ä p < 1, the Sobolev spaces Wk;p.˝I Rm/ are separable, and if 1

22 k;p m k;p This means that W .˝I R / WD f .u1;:::;um/ j ui 2 W .˝/; i D 1;:::;m g. 600 B Elements of Measure Theory and Function Spaces

Hk.˝I Rm/ WD Wk;2.˝I Rm/: (B.4.4)

To make traces well deﬁned on the boundary WD @˝ WD ˝N n ˝ with ˝N WD cl.˝/, we must qualify ˝ suitably. We say that ˝ is a domain of Ck-class if there is a ﬁnite number of overlapping parts i of the boundary of that are graphs of Lipschitz Ck-functions in local coordinate systems and ˝ lies on one side of .23 Replacing Ck with C0;1, we say that ˝ is of the C0;1-class or that it is a Lipschitz domain. Theorem B.4.1 (Sobolev embedding [566]). The continuous embedding

W1;p.˝/ Lp .˝/ (B.4.5) holds if the so-called Sobolev exponent p is deﬁned as 8 ˆ dp <ˆ for p < d; d p p WD ˆ an arbitrarily large real for p D d; (B.4.6) :ˆ 1 for p > d:

Moreover, for p > d, we have also W1;p.˝/ C.˝/N .24 Theorem B.4.2 (Rellich, Kondrachov [312, 501]25). The compact embedding

W1;p.˝/ b Lp .˝/ ; 2 .0; p1; (B.4.7) holds for p from (B.4.6). Moreover, for p > d, we have also W1;p.˝/ b C.˝/N . Reiterating Theorems B.4.1 and B.4.2, one gets the following corollary26

23 Written formally, we require the existence of transformation unitary matrices Ai and open sets d1 0;1 d1 d Gi 2 R and gi 2 C .R / such that each i can be expressed as i DfAij 2 R ; d .1;:::;d1/ 2 Gi;d D gi.1;:::;d1/g and fAij 2 R ;.1;:::;d1/ 2 Gi; d gi.1;:::;d1/ "0. 24It is to be understood that each u 2 W1;p.˝/ admits a continuous extension on the closure ˝N or ˝. 25The pioneering work of Rellich dealt with p D 2 only. 26E.g., W2;p.˝/ W1;dp=.dp/.˝/ by applying Theorem B.4.1 to ﬁrst derivatives, and applying Theorem B.4.1 once again for dp=.d p/ instead of p, one arrives at W1;dp=.dp/.˝/ Ldp=.d2p/.˝/, provided 2p < d. Repeating once again yields W3;p.˝/ b Ldp=.d3p/.˝/, provided 3p < d,etc. B Elements of Measure Theory and Function Spaces 601

Corollary B.4.3 (Higher-order Sobolev embedding). (i) If kp < d, the continuous embedding Wk;p.˝/ Ldp=.dkp/.˝/ and the compact embedding Wk;p.˝/ b Ldp=.dkp/.˝/ hold for every >0. (ii) For kp D d, we have Wk;p.˝/ b Lq.˝/ for every q < 1. (iii) For kp > d, we have Wk;p.˝/ b C.˝/N . Theorem B.4.4 (Trace operator.27). There is exactly one linear continuous opera- 1;p 1 tor T W W .˝/ ! L . / such that for every u 2 C.˝/N , we have Tu D uj (= the restriction of u on ). Moreover, T remains continuous (respectively is compact) as the mappings

1;p p] u 7! uj W W .˝/ ! L . /; respectively (B.4.8a)

1;p p] ] u 7! uj W W .˝/ ! L . / ; 2 .0; p 1; (B.4.8b) provided the so-called Sobolev trace exponent p] is deﬁned as 8 dp p <ˆ for p < d ; ] d p p WD ; (B.4.9) :ˆ an arbitrarily large real for p D d 1 for p > d :

We call the operator T from Theorem B.4.4 the trace operator, and write simply 1;p 1;p uj instead of Tu even if u 2 W .˝/ n C.˝/N . Then we define W0 .˝/ WD 1;p k;p f v 2 W .˝/ j vj D 0 g.Fork >1, we define similarly W0 .˝/ WD f v 2 k;p i ki;p di W .˝/ jrv 2W0 .˝I R /; i D 0;:::;k1 g. Assuming ˝ a Lipschitz domain, we denote by D .x/ 2 Rd the unit outward normal to the boundary at a point x 2 ; this is well defined H d1-almost everywhere on .28 The multidimensional integration by parts Z Z @ @v Á z d1 v C z dx D vz i dS with dS WD H j (B.4.10) ˝ @xi @xi

0 holds for every v 2 W1;p.˝/ and z 2 W1;p .˝/ and for all i D 1; ::; d. Considering . ;:::; / z D z1 zd , writingP (B.4.10)forzi instead of z, and ﬁnally summing over d @ i D 1;:::;d with div z WD 1 zi the divergence of the vector ﬁeld z,wearrive iD @xi at a formula that we will often use:

27 1;p 11=p;p In fact, u 7! uj W W .˝/ ! W . /, where the Sobolev–Slobodecki˘ı space W11=p;p. / deﬁned now on a .d1/-dimensional manifold instead of a d-dimensional ] domain ˝. Then, similarly as in Theorem B.4.1, we have the embedding W11=p;p. / Lp . /, ] respectively W11=p;p. / b Lp . /. 28This normal can be deﬁned by means of gradients of Lipschitz functions describing locally as their graphs. By Rademacher’s theorem, these derivatives exist H d1-almost everywhere on . 602 B Elements of Measure Theory and Function Spaces

Theorem B.4.5 (Green formula [233]29). The following formula holds for every 0 v 2W1;p.˝/ and z2W1;p .˝I Rd/: Z Z v.div z/ C zrv dx D v.z / dS: (B.4.11) ˝

A variant of the Green formula on a curved smooth surface is sometimes useful: Considering a scalar field in the neighborhood of , let us define the surface . / I gradient rSg WD rg P with the projector P D ˝ onto a tangent space. Alternatively, pursuing the concept of fields defined exclusively on , we can consider g W ! R and extend it to a neighborhood of and then again define . / rSg WD rg P, which, in fact, does not depend on the particular extension. Then, v Rd for a vector field W ! , we define the surface divergence, denoted by divS, . / . v/ v v as divSg WD tr rSg . With such definitions, we have divS g D gdivS C rSg;cf. e.g., [210, Formula (21)] for the vectorial case. Integrating over yields the Green formula on the surface : Z Z Z Z v v . v/ . v/ ; g divS dS C rSg dS D divS g dS D g ds (B.4.12) @ where here is the unit outward normal to the .d2/-dimensional boundary @ of . 1/ the d -dimensional surface .Cf.also[484, 600]. The operator divSrS is called the Laplace–Beltrami operator. A generalization of Sobolev spaces for k 0 noninteger is often useful for various finer investigations: We define the Sobolev–Slobodecki˘ı space as Wk;p.˝/ WD Œk;p.˝/ < f u2W jkukWk;p.˝/ 1gwith ˇ Z ˇ jrŒku.x/ rŒku./jp Wk;p.˝/ WD u2WŒk;p.˝/ ˇ dxd<1 ; nCp.kŒk/ ˝˝ jx j (B.4.13) where Œk denotes the integer part of k. They are Banach spaces. In fact, Corol- lary B.4.3 holds also for k 0 noninteger.

B.5 Abstract functions on Œ0; T: their variations, integrals, derivatives

We now deﬁne spaces of abstract functions on a bounded interval Œ0; T R valued in a Banach space V , invented by Bochner [75]. We say that z W Œ0; T ! V

R R 29 v v v du Putting z Dru into (B.4.11), we get ˝ u Cr ru dx D d dS derived in [233]. In fact, (B.4.11) holds, by continuous extension, under weaker assumptions; cf. [446]. B Elements of Measure Theory and Function Spaces 603

v 1.v / is simple if it takes only a finiteR number of valuesP i 2 V and Ai WD z i T . / L 1. /v is Lebesgue measurable; then 0 z t dt WD finite Ai i. We say that z W .0; T/ ! V is Bochner measurable if it is the pointwise limit (in the strong topology) of V of a sequence fzkgk2N of simple functions; i.e., zk.t/ ! z.t/ for a.a. t 2 Œ0; T. The space of all bounded (everywhere defined) Bochner measurable mappings z W Œ0; T ! V is denoted by B.Œ0; TI V /. It is a linear space under pointwise multiplication/addition, and if equipped with the norm kzkB.Œ0;TIV / D . / 30 sup0ÄtÄT kz t kV , also a Banach space. Moreover, we say that z W Œ0; T ! V is weakly measurable if hv; z./i is Lebesgue measurable for every v 2 V .IfV D .V0/ for some Banach space V0, then z W Œ0; T ! V is weakly* measurable if hv; z./i is Lebesgue measurable for every v 2 V 0. For y subset I R, let us denote by F.I/ the collection of all finite subsets 31 .t1;:::;tn/ I, n 2 N, considered ordered and satisfying t1 < t2 < ::: < tn. A variation of z W I ! V with respect to the norm of V is defined as

Xn . ; / . / . 1/ : Var V z I WD sup z ti z ti V (B.5.1) . ;:::; / F. / t1 tn 2 I iD1

Realizing that F.I/ is a directed set if ordered by inclusion, the “sup” in (B.5.1) can be replaced by “lim.” A subspace of mappings z 2 B.II V / with bounded variation Var V .z; I/<1 is a Banach space if normed by kkB.IIV / C Var V .; I/, denoted by BV.II V /.32 It should be pointed out that for I a closed interval, BV.˝/ for ˝ D int I as deﬁned in (4.2.98) on p. 306, ignores values on zero-measure sets and is thus obviously different from BV.I/ as deﬁned here. 33 We say that z W Œ0;PT ! V is absolutely continuous if for each ">0, there ı>0 K . / . / " is such that kD1 kz tkP z sk kV Ä whenever tk1 Ä sk Ä tk Ä T for 1;:::; N 0 K ı k D K 2 , t0 D , and kD1 tksk Ä . The space of absolutely continuous mappings z W Œ0; T ! V is denoted by AC.Œ0; TI V /. Always AC.Œ0; TI V / BV.Œ0; TI V /. .0; / Œ0; A pointRt 2 T is called a Lebesgue point of z W T ! V if 1 h=2 lim C kz.tC#/ z.t/kd# D 0. Analogously, a right Lebesgue point h!0 h h=2 R 1 h Œ0; / C . #/ . / # 0 t2 T means limh!0 h 0 kz tC z t kd D .

30 Note that metrizability allows us to work with sequences and that every Cauchy sequence .zk/k2N in B.Œ0; TI V / induces sequences .zk.t//k2N that are Cauchy in V , and their limit z.t/ also forms the limit u in B.Œ0; TI V /, which is attainable by a sequence of simple functions, because each zk is simple; here a diagonalization procedure applies. 31In particular, for I D Œr; s,thenPart.Œr; s/ F.Œr; s/. 32 . / R If I is a closed interval, in view of (B.5.6) below, we can also write kzkBV.IIV / D supt2I kz t kV C k k dz.t/. I V 33If V D R1, this deﬁnition naturally coincides with absolute continuity with respect to the Lebesgue measure on Œ0; T as deﬁned on p. 595. 604 B Elements of Measure Theory and Function Spaces

Theorem B.5.1 (Pettis [476].34). If V is separable, then u is Bochner measurable if and only if it is weakly measurable. Œ0; Considering simple functionsR fukgk2N as above, we call z W R T ! V T . / . / 0 T . / BochnerR integrable if limk!1 0 kz t zk t kV dt D . Then 0 z t dt WD T limk!1 0 zk.t/ dt; this limit exists and is independent of the particular choice of the sequence .zk/k2N. Moreover, if V is separable, then a Bochner . / measurable function z is BochnerR integrableR if and only if kz kV is Lebesgue integrable. Then also k T z.t/ dtk Ä T kz.t/k dt. From this, we can R 0 V 0 V 1 h=2 C . #/ # . / see that limh!0 h h=2 z tC d ! z t at each Lebesgue point t 2 R =2 R =2 .0; T/; note that kz.t/ 1 h z.tC#/d#k Dk1 h z.tC#/z.t/d#k Ä R h h=2 V h h=2 V 1 h=2 . #/ . / # h h=2 kz tC z t kV d . Theorem B.5.2. If z2L1.0; TI X/, then a.e. t2.0; T/ is a Lebesgue point for z. An analogous assertion holds for right Lebesgue points. Every u 2 Cw.Œ0; TI V / WD fŒ0; T ! V weakly continuousg is an Rexample of a Bochner integrable function. For such u, the LebesgueR integral T T 0 u.t/ dt can alternatively be deﬁned like a Riemann integral, i.e., 0 u.t/ dt D lim¿.˘/!0 Riem.u;˘/over all partitions ˘ of Œ0; T the form 0 D t0 < t1 <:::< tN1 < tN D T, N 2 N, with ¿.˘/ WD maxjD1;:::;N .tj tj1/, and where

XN Riem.u;˘/WD u.tj/.tjtj1/ (B.5.2) jD1 R T is a Riemann sum for the integral 0 u.t/ dt with respect to the partition ˘ of Œ0; T. This cannot hold for a general u 2 L1.0; TI V /, which is deﬁned only a.e. on .0; T/. Anyhow, the following result still holds: Theorem B.5.3 (Approximation by Riemann sums). 35 For every u 2 L1.0; TI V / ˘ m ¿.˘ m/ 0 with V a Banach space, thereR exists a sequence of partitions with ! m T such that Riem.u;˘ / ! 0 u.t/ dtinV for m !1. . ;˘/ A generalization of the construction lim¿.˘/!0 Riem u yields the Riemann–. Stieltjes integral of a scalar-valued function u with respect to a measure v induced

34In fact, [476] works with a general Banach space, showing the equivalence of the Bochner measurable mappings with a.e. separably valued weakly measurable mappings. 35See [149, Sect.4.4]. In the scalar-valued variant, this sort of result dates back to Hahn [252], recently also to be found, e.g., in [372, Sect.A.3]. Moreover, the assertion holds in fact for a.a. sequences of partitions ˘ m with ¿.˘ m/ ! 0. Note that we rely on the fact that u is deﬁned everywhere on Œ0; T, and in particular, that u.T/ is deﬁned, although its particular value is not important, because the term u.tN /.tN tN1/ can always be made arbitrarily small by sending tN1 ! tN D T. B Elements of Measure Theory and Function Spaces 605

R R T R v T u v.t/ u v.t/ T u v.t/ by a nondecreasing function , defined by 0 d WD 0 d D 0 d if there is equality of the upper and lower Riemann–Stieltjes integrals. These two integrals are defined respectively by the infimum and supremum of the upper and lower Darboux sums as Z T XN u dv.t/ WD inf Darb.u;˘;v/; Darb.u;˘;v/WD sup u.t/ v.tj/v.tj1/ ; 0 ˘ Œ ; jD1 t2 tj1 tj (B.5.3) Z T XN u dv.t/ WD sup Darb.u;˘;v/; Darb.u;˘;v/WD inf u.t/ v.tj/v.tj1/ : ˘ t2Œt 1;t 0 jD1 j j (B.5.4) for ˘ again ranging all partitions of Œ0; T of the form 0 D t0 < t1 < ::: < tN1 < tN D T, N 2 N;cf.[554, Chap. 6] for details. Notably, (B.5.4) bears a straightforward generalization for v valued in a Banach-space V and f .t; / a (possibly even nonlinear) 1-homogeneous convex functional on V : Z T XN Á v.tj/v.tj1/ f .t; / dv.t/WD lim sup .tjtj1/ inf f t; ˘ F.Œ0; / t2Œt 1;t t t 1 0 2 T jD1 j j j j by 1-homo- geneity D lim sup Darb.f ;˘;v/ ˘2F.Œ0;T/ XN with Darb.f ;˘;v/WD inf f t;v.tj/v.tj1/ : t2Œt 1;t jD1 j j (B.5.5)

Here, as f depends on t, the mapping ˘ 7! Darb.f ;˘;v/is not monotone and considering its supremum like in (B.5.4) would not work properly. Therefore, in (B.5.5)wehaveusedthatF.Œ0; T/ is a directed set when ordered by the inclusion, and thus “limsup” could be used instead. This is a so-called Moore-Pollard’s v modification of the Riemann-StieltjesR . construction,R cf.. [434, 490]. If is absolutely continuous, (B.5.5) equals T Œf .t; /v.t/ dt D T f .t; v.t// dt, but let us emphasize 0 0 . that in the general non-absolutely-continuous case, v does not need to be valued in V or even defined at all for (B.5.5). Only if f is non-negative and independent of t, one can use the supremum (or, in fact, also the Moore-Smith limit), which is the Riemann-Stieltjes-type construction used for the definition (3.2.12) on p. 124. By this definition, we can also write VarV from (B.5.1) in terms of the lower Riemann– Stieltjes integral simply as Z T ;Œ0; . /: Var V z T D V dz t (B.5.6) 0 606 B Elements of Measure Theory and Function Spaces

In the special case V D Rd equipped with the Euclidean norm, the construction (B.5.6) based on the lower Riemann–Stieltjes integral is known as the length of the curve z W Œ0; T ! Rd,cf.[554, Chap. 6], and (B.5.5) can thus be understood as a generalization of this construction. An important attribute is the additivity of (this generalization of) the lower Riemann–Stieltjes integral, i.e.,36 Z Z Z t2 t3 t2 8 0 Ä t1 Ä t3 Ä t2 Ä T W f .t; / dv.t/ D f .t; / dv.t/ C f .t; / dv.t/: t1 t1 t3 (B.5.7)

Also, if f W Œ0; TV ! R1 is lower semicontinuous with respect to a topology T on V , then lower semicontinuity is an important attribute of this deﬁnition37: Á Z Z T T T 8 t2Œ0; T W vk.t/ ! v.t/ ) lim inf f .t; / dvk.t/ f .t; / dv.t/: k!1 0 0 (B.5.8)

In general if the “integrator” function v is vector-valued as in (B.5.5)butthe functional f is now linear (being in duality with values of v), then neither non- negativity of f nor the monotonicity of v are relevant and again ˘ 7! Darb.f ;˘;v/ is not monotone. Then (B.5.4) must be modiﬁed by replacing “sup” with “lim sup” as we did already in (B.5.5) which now takes the form: Z ˝T ˛ f .t/; dv.t/ WD lim sup Darb.f ;˘;v/ 0 ˘2F.Œ0;T/ XN ˝ ˛ where now Darb.f ;˘;v/WD inf f .t/; v.tj/v.tj1/ : (B.5.9) t2Œt 1;t jD1 j j

This limit-construction is a special case what is called a (here lower) Moore-Pollard- Stieltjes integral,cf.[434, 490]. In fact, it can be understood as a very special case of a so-called multilinear Stieltjes integral used here for two vector-valued functions in duality. Like in the classical scalar situation of lower Riemann-Stieltjes integral (B.5.4), the sub-additivity of the integral with respect to u and to v holds together with the additivity with respect to the domain like (B.5.7) holds. For 1 Ä p < 1,aBochner space Lp.0; TI V / is the linear space (of classes with respect to equivalence a.e.) of Bochner integrable functions z W .0; T/ ! V

36The inequality in (B.5.7) is a simple consequence of the definition (B.5.5). The opposite inequalityR relies on the fact that a refinement of a partition of Œt1; t2 by including ft3g still approxi- t2 . ; / v. / . ;˘;v/ mates f t d t from below, and then on the inequality sup˘2 .Œ 1; 2/; ˘3 3 Darb f Ä t1 Part t t t sup Darb.f ;˘;v/C sup Darb.f ;˘;v/; cf. also (B.5.6). ˘2Part.Œt1;t3/ ˘2Part.Œt3;t2/ R 37 T !1 . ; / v . / Thanks to the joint lower semicontinuity of f , we have lim infk 0 f t d k t lim infk!1 Darb.f ;˘;vk/ Darb.f ;˘;v/ for every ˘ fixed, and taking the supremum over ˘’s yields (B.5.8); cf. also the arguments (5.1.106)–(5.1.107) for more details. B Elements of Measure Theory and Function Spaces 607 R T . / p < satisfying 0 kz t kV dt 1. This space is a Banach space if endowed with the norm 8 Z Á1= ˆ T p p < z.t/ dt if 1 Ä p < 1; z WD 0 V (B.5.10) Lp.0;TIV / :ˆ . / : ess sup z t V if p D1 t2I

If V has a predual, i.e., V D .V 0/ for some Banach space V 0, the notation p Lw.0; TI V / stands for the space of weakly* measurable p-integrable (or if p D1, essentially bounded) functions .0; T/ ! V . We often use an equidistant partition of .0; T/ into subintervals of length WD 2KT, K 2N. Proposition B.5.4 (Uniform convexity). If V is uniformly convex and 1

Thus, if p 2 .1; 1/ and V is reflexive and separable, then Lp.0; TI V / is reflexive. 1 .0; / 1.0; 0/ (ii) Moreover, Lw TI V is dual to the space L TI X . 1 .0; / 1.0; / Let us emphasize that in general, Lw TI V is not equal to L TI V .If 1.0; / 1 .0; / V is separable reflexive, then L TI V D Lw TI V by Pettis’s theorem (Theorem B.5.1). Considering Banach spaces V 0, V 1,..., V k and a W IV 1V k ! V 0,let us define the Nemytski˘ı mappings Na again by the formula (B.3.11). The following generalization of Theorem B.3.9 holds:

Theorem B.5.6 (Nemytski˘ı mappings on Bochner spaces [364]). Let V 0, V 1,...,V k be separable Banach spaces, a W Œ0; TV 1V k ! V 0 a Carathéodory mapping,38 and suppose that the growth condition

Xk = pi p0 p0 a.x; r1 :::;rk/ Ä .x/ C C ri for some 2L .0; T/; V 0 V i iD1 (B.5.12)

38Generalizing the ﬁnite-dimensional case, this means that for a.a. t 2 Œ0; T, a.t; / W V 1 V k ! V 0 is to be (norm, norm)-continuous and a. ; r1;:::;rk/ W Œ0; T ! V 0 is to be measurable. 608 B Elements of Measure Theory and Function Spaces

p1 pk holds with p0; p1;:::;pk as in (B.3.12). Then Na maps L .0; TI V 1/L p0 .0; TI V k/ continuously into L .0; TI V 0/. . /. We denote by. the distributional derivative of u understood as the abstract linear operator u 2 Lin.D.Œ0; T/; .V ; weak// defined by Z . T . 8 ' 2 D.0; T/ W u.'/ WD u ' dt; (B.5.13) 0 where again D.0; T/ stands for smooth compactly supported functions on .0; T/. Then we define the Sobolev–Bochner space W1;p.0; TI V / by ˇ 1; 1 ˇ . W p.0; TI V / WD z 2 L .0; TI V / ˇ z 2 Lp.0; TI V / : (B.5.14) . It is a Banach space if normed by kzkW1;p.0;TIV / WD kzkL1.0;TIV / CkzkLp.0;TIV /. The abstract setting for evolution problems often relies on the construction of a so-called evolution (also called Gelfand’s) triple. Assuming that V is embedded continuously and densely into a Hilbert space H identified with its own dual H Á H ,wehavealsoH V continuously. Indeed, denoting by i W V ! H the mentioned embedding, we have that the adjoint mapping i (which is continuous) maps H into V and is injective, because i is just the restriction of linear continuous functionals H ! R on the subset V , and different continuous functional must remain different when restricted to such a dense subset. The mentioned identification of H with its own dual H yields altogether

V H Á H V I (B.5.15) the triple .V ; H; V / is called a Gelfand triple. The duality pairing between V and V is then a continuous extension of the inner product on H, denoted by .j/H , i.e., for u 2 H and v 2 V,wehave39 ˝ ˛ ˝ ˛ ˝ ˛ ˝ ˛ v ;v ; v ;v ;v : uj H D u H H D u i H H D i u V V D u V V (B.5.16) Moreover, the embedding H V is dense. Lemma B.5.7 (Integration by parts formula). Let V H Š H V . 0 Then Lp.0; TI V / \ W1;p .0; TI V / C.0; TI H/ continuously, and the following 0 integration by parts formula holds for every u;v 2 Lp.0; TI V / \ W1;p .0; TI V / and 0 Ä t1 Ä t2 Ä T: Z D E D E t2 du dv u.t2/jv.t2/ u.t1/jv.t1/ D ;v.t/ C u.t/; dt: (B.5.17) t1 dt dt

39The equalities in (B.5.16) follow successively from the identification of H with H ,the embedding V H, the definition of the adjoint operator i, and the identification of iu with u. B Elements of Measure Theory and Function Spaces 609

In particular, the formula (B.5.17)foruD v gives Z t2 D E 1 2 1 2 du u.t2/ u.t1/ D ; u.t/ dt ; (B.5.18) 2 H 2 H t1 dt

1 . / 2 and shows that the function t 7! 2 ku t kH is absolutely continuous and its derivative exists a.e. on .0; T/ and D E 1 d 2 du u.t/ D ; u.t/ for a.a. t2.0; T/: (B.5.19) 2 dt H dt Another important ingredient often used in evolution problems deals with compactness:

Lemma B.5.8 (Aubin and Lions [29, 361]). Let V 1, V 2, V 3 be Banach spaces, V 1 separable and reﬂexive, V 1 b V 2 (a compact embedding), V 2 V 3 (a p continuous embedding), 1

p L .0; TI V 1/ \ BV.Œ0; TI V 3/ n ˇ o p ˇ . p D u2L .0; TI V 1/ ˇ u2M .Œ0; TI V 3/ b L .0; TI V 2/ (B.5.20)

p in the sense that bounded sets in L .0; TI V 1/ \ BV.Œ0; TI V 3/ are sequentially p relatively compact in L .0; TI V 2/. In particular, Lemma B.5.9 combined with Proposition B.3.1(ii)–(iii) yields that p every sequence bounded in L .0; TI V 1/ \ BV.Œ0; TI V 3/ possesses a subsequence 42 converging a.e. in .0; T/ strongly in V 2. There is another selection principle

40This means that every bounded sequence is mapped to a relatively compact one. In fact, these original references deal with a slightly different compactness concept and slightly stronger assumptions, e.g., reflexivity of V 3 or 1probability theory for distribution functions, states that each bounded sequence of nondecreasing bounded functions on an interval possesses a subsequence that converges pointwise to a nondecreasing limit function. 44 Corollary B.5.11. Let V be as in Theorem B.5.10, and let V 1 be a reflexive Banach space continuously embedded into V and having a separable predual. Every bounded sequence in Cw.Œ0; TI V 0/ \ BV.Œ0; TI V / contains a subsequence converging weakly in V 1 everywhere on Œ0; T, and the limit lives in B.Œ0; TI V 0/ \ BV.Œ0; TI V /. A combination of Helly’s principle and an Arzelà–Ascoli-type modification of the Aubin–Lions lemma (Lemma B.5.8) allows us to make the following assertion, applicable to discontinuous piecewise constant functions arising in time discretizations:

Corollary B.5.12. Assuming again V 1 b V 2 V 3 with V 1 and V 3 reﬂexive, and 10, we have 1 fuN g >0 bounded in B.Œ0; TI V 1/ uN ! u strongly in L .0; TI V 2/ ) . 1;q.0; / fu g >0 bounded in W TI V 3 for a subsequence (B.5.21) for a piecewise constant uN and the corresponding piecewise afﬁne continuous u , 1 1;q and for some u 2 L .0; TI V 2/ \ W .0; TI V 3/. 45 1;q Proof. Considering a bounded sequence .uN / >0 in B.Œ0; TI V 1/\W .0; TI V 3/ 1;q and taking into account W .0; TI V 3/ BV.Œ0; TI V 3/, by Helly’s selection principle, we can be certain that uN .t/ ! u.t/ weakly in V 3 for every t 2 Œ0; T. Then also uN .t/ ! u.t/ weakly in V 1, and by the compact embedding

43See [55, 432] for separable reﬂexive V ,or[143, Lemma 7.2] for the general case. 44 1 By Cw.Œ0; TI V 0/ L .0; TI V 0/, there is a subsequence converging weakly* in 1 L .0; TI V 0/, and in particular, the limit is Bochner measurable. By Theorem B.5.10, selecting a further subsequence, we get the pointwise convergence weakly in V . At each particular time instance, its subsequences must converge in V 0, but their limits must be again the same as the limit in V . 45This proof paraphrases the arguments of the proof of [532, Lemma 7.10], claiming the compact 1;q embedding Cweak.Œ0; TI V 1/ \ W .0; TI V 3/ b C.Œ0; TI V 2/, which, however, cannot be used directly, because uN 62 Cweak.Œ0; TI V 1/. B Elements of Measure Theory and Function Spaces 611

V 1 b V 2,alsouN .t/ ! u.t/ strongly in V 2 for every t 2 Œ0; T. The sequence fuN W Œ0; T ! V 3g >0 is “equicontinuous” (although particular mappings uN are not continuous) because Z Z t2 t2 . . u .t1/u .t2/ Ä u dt Ä 1 u dt N N V 3 V 3 t1 V 3 t1 . 11=q. Äk1k q0 u Djt1t2j u L .Œt1;t2/ Lq.0;TIV 3/ Lq.0;TIV 3/ for every 0 Ä t1 < t2 Ä T. Assume that the selected sequence fuN g >0 does 1.0; / 1 >0 not converge to u in L TI V 2 . Thus kuN ukL .0;TIV 2/ for some and for all >0(from the already selected subsequence), and we would get kuN .t /u.t /kV 2 for some t . By compactness of Œ0; T, we can further select a subsequence and some t 2 Œ0; T so that t ! t. Then we have u.t / ! u.t/ in V 2. By the above proved equicontinuity, we have also uN .t / ! u.t/ weakly in V 3.By the boundedness of fuN .t /g >0 in V 1 b V 2,wehavealsouN .t / ! u.t/ in V 2.

Then kuN .t /u.t /kV 2 !ku.t/u.t/kV 2 D 0, a contradiction. Thus (B.5.21)is proved. ut There are further generalizations of Theorem B.5.10 that do not need any linear structure of V and rely fully on a metric. In [373, Thm. 3.2], a version was provided for general quasimetrics as introduced in Chapter 2.46 We provide a very general version requiring that the underlying space Z be only a Hausdorff topological space and, instead of a single quasimetric, using a sequence .Dk/k2N of quasidistances and a limit D1, all deﬁned on Z, and then showing that Theorem 2.1.24 can be deduced from that. In this appendix, we explicitly state at each instance that all topological notions such as semicontinuity and compactness are meant in the “sequential sense.” Speciﬁcally, we assume

8 k 2 N1 8 z1; z2; z3 2 Z W

Dk.z1; z2/ D 0 ” z1 D z2; (B.5.22a)

Dk.z1; z3/ Ä Dk.z1; z2/ C Dk.z2; z3/I (B.5.22b)

zk ! z and Qzk ! QzIH)D1.z; Qz/ Ä lim inf Dk.zk; Qzk/: (B.5.22c) k!1 D1 W ZZ ! Œ0; 1 is sequentially lower semicontinuous: (B.5.22d)

Theorem B.5.13 (General Helly selection principle). Assume that the sequence

.Dk/k2N1 satisﬁes the conditions (B.5.22). Moreover, let K be a sequentially compact subset of Z, and zk W Œ0; T ! Z; k 2 N, a sequence satisfying

8 t2Œ0; T 8 k2N W zk.t/ 2 K; (B.5.23a) . Œ0; / < : sup DissDk zkI T 1 (B.5.23b) k2N

46Various generalizations of this kind have been made in [64, 121] and, assuming, in addition, continuity over Œ0; T,alsoin[119, 120, 122]. 612 B Elements of Measure Theory and Function Spaces

. / Œ0; Z Then there exist a subsequence zkl l2N, a limit function z W T ! , and a nondecreasing function ı W Œ0; T ! Œ0; 1Œ with the following properties: 8 t 2 Œ0; T W ı.t/ D lim DissD z I Œ0; t ; (B.5.24a) l!1 kl kl Œ0; . / Z . / . / K; 8 t 2 T W zkl t ! z t and z t 2 (B.5.24b) 8 s; t2Œ0; T with s

. ;Œ ; / . ;Œ ; / Proof. Let us abbreviate Dissk z s t WD DissDk z s t .Ofcourse,wehave Dk.z.s/; z.t// Ä Dissk.zI Œs; t/. We define the functions dk W Œ0; T ! Œ0; 1 with dk.t/ D Dissk.zkI Œ0; t/, which are nondecreasing by definition and uniformly bounded by (B.5.23b). Hence, the classical Helly’s selection principle [264] for real-valued functions provides a subsequence such that d .t/ ! ı.t/ for all t 2 Œ0; T. Hence, ı W Œ0; T ! Œ0; 1 is Qk n also nondecreasing and bounded. This proves (B.5.24a). Denote by IJ Œ0; T the set of discontinuity points of ı. Then IJ is countable. Hence, we may choose a countable, dense subset I0 of Œ0; T with IJ I0.For each t 2 I0, every subsequence of .z .t// N lies in the sequentially compact set Qk n n2 K Z and thus contains a convergent subsequence. Using Cantor’s diagonalization procedure, we find a subsequence .z / N of .z / N such that (B.5.24a) remains kl l2 Qk n n2 true, and additionally, we have

. / Z . / : 8 t 2 I0 W zkl t ! z t for l !1

This defines the limit function z W I0 ! Z. To prove convergence on Œ0; T n I0, we use the continuity of ı.Wefixt 2 Z Œ0; T n I0. Then the sequence .zk .t//l2N has a convergent subsequence z .t/ ! l Okm z. Moreover, there exists a sequence tn 2 I0 with tn ! t. Below, we show that . / Z Z . . // z tn ! z. By the Hausdorff property of , we conclude that zkl t l2N has exactly one accumulation point, and we define z.t/ D z. Z Z To prove z.tn/ ! z, first note that we may assume z.tn/ ! Oz 2 K, since each z.tn/ lies in the sequentially compact set K. Next we consider tn < t. Then, using (B.5.22c), we have

D1.z.tn/; z/ Ä lim inf D .z .tn/; z .t// m!1 Okm Okm Okm

Ä lim inf Diss .z I Œtn; t/ D ı.t/ı.tn/: (B.5.25) m!1 Okm Okm

Similarly, for t < tn, we obtain D1.z; z.tn// Ä ı.tn/ ı.t/. Using the continuity of ı in t, we conclude that ˚ « ˇ ˇ ˇ ˇ min D1.z.tn/; z/; D1.z; z.tn// Ä ı.t/ı.tn/ ! 0 for n !1: (B.5.26) B Elements of Measure Theory and Function Spaces 613

Employing (B.5.22d), we obtain ˚ « ˚ « min D1.z; Oz/; D1.Oz; z/ Ä lim inf min D1.z; z.tn//; D1.z.tn/; z/ D 0: n!1

Z Thus, using (B.5.22a), we conclude that Oz D z, and hence z.tn/ ! z. Assertion (B.5.24b) is proved. The ﬁnal estimate is obtained using (B.5.22c) again. For every partition s Ä t0 < t1 <:::

XN XN D1.z.tj1/; z.tj// Ä lim inf Dk .zk .tj1/; zk .tj// l!1 l l l jD1 jD1 XN Ä lim inf Dk .zk .tj1/; zk .tj// l!1 l l l jD1

Ä lim inf Dissk .zk I Œs; t/ D ı.t/ı.s/: (B.5.27) l!1 l l

Thus, Diss1.zI Œs; t/ Ä ı.t/ı.s/, and (B.5.24c) is proved. ut Note that (B.5.22a,b) corresponds to (D1), and (B.5.22c,d) reduces to (D2) in the case Dk D D. Note also that Theorem B.5.13 reduces to Theorem 2.1.24 if we assume that Dk D D for all k 2 N1. Appendix C Young Measures and Beyond

Here we present brieﬂy an analytical tool that yields, roughly speaking, (local) compactiﬁcations of function spaces that are also convex in a natural way, imitating thereby the most important topological/geometric properties of Euclidean spaces, and that allow for continuous extensions of Nemytski˘ı mappings.

C.1 Young measures: a tool to handle oscillations

Let us begin with the simplest situation: ˝ Rd a domain S be a metrizable compact set and U WD fu W ˝ ! S measurable g. Let us now embed U into the dual space L1.˝I C.S// by means of the embedding i W U ! L1.˝I C.S// deﬁned by Z hi.u/; hiWD h.x; u.x// dx : (C.1.1) ˝

The weak* closure of i.U/ in L1.˝I C.S// is denoted by Y.˝I S/, i.e.,1 n ˇ o 1 ˇ Y.˝I S/ WD 2 L .˝I C.S// ˇ 9.uk/k2N U W D w*-lim i.uk/ : (C.1.2) k!1

1Note that in (C.1.2), we conﬁned ourselves to considering limits of sequences, relying on the fact that L1.˝I C.S// is separable because S is metrizable and compact.

© Springer Science+Business Media New York 2015 615 A. Mielke, T. Roubícek,ˇ Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences 193, DOI 10.1007/978-1-4939-2706-7 616 C Young Measures and Beyond

1 .˝ / For X a Banach space, let Lw I X denote the Banach space of weakly* measurable mappings W ˝ ! X, i.e., all functions x 7!h .x/; 'i are Lebesgue measurable for every ' 2 X. There is an isometric isomorphism W 1.˝ . // 1 .˝ M . // L I C S ! Lw I S deﬁned as

1 .˝ M . // 1.˝ . // W 7! W Lw I S ! L I C S (C.1.3) with 2 L1.˝I C.S// deﬁned by Z Z Z

h; hiWD Œh .x/ dx D h.x; s/ x.ds/ dx; (C.1.4) ˝ ˝ S wherewehavealsousedh W ˝ ! R deﬁned, for a.a. x 2 ˝,by Z

Œh .x/ WD h.x; s/ x.ds/: (C.1.5) S

We can use this isometric isomorphism for an alternative and perhaps “optically” more explicit construction. Let us deﬁne the embedding ı W U ! 1 .˝ M . // Lw I S by means of

Œı.u/.x/ WD ıu.x/ ; (C.1.6)

C where ıs 2 M0 .S/ for s 2 S denotes the Dirac distribution supported at s. Let us denote the set of all Young measures by Y .˝I S/, deﬁned by n ˇ o ˇ Y .˝ / 1 .˝ M . // ˇ M C. / ˝ : I S WD Df xgx2˝ 2Lw I S x 2 0 S for a.a. x2 (C.1.7)

1 .˝ M . // 1.˝ . // The mapping W Lw I S ! L I C S from (C.1.3) maps the set of all Young measures Y .˝I S/ onto Y.˝I S/ and ı i D ı. Theorem C.1.1. 2 The set of Young measures Y .˝I S/ is convex and weakly* 1 .˝ M . // ı. / Y .˝ / compact in Lw I S , and U is weakly* dense in I S , and therefore, the convex compactiﬁcations .Y .˝I S/; ı/ and .Y.˝I S/; i/ of U are equivalent via the afﬁne homeomorphism from (C.1.3).

We see that alternatively, each parameterized measure Df xgx2˝ can be considered a linear continuousR functional on a suitable space of integrands, given by the formula h 7! ˝ Œh .x/ dx, i.e.,

2See, e.g., [520]. C Young Measures and Beyond 617 Z Z

W h 7! h.x; s/ x.ds/ dx : (C.1.8) ˝ S

This is basically the original understanding in the work of L.C. Young.3 Parameter- ized probability measures contain enough information to describe the L1.˝I Rm/- . / ˝ Rm weak* limits of h ı uk k2N for h W S ! ; namely, we have h ı uk ! h weakly* if h is a bounded Carathéodory mapping. In other words, the Nemytski˘ı mapping Nh induced by h has a weak*-continuous extension to NN h W 7! h . 1 m Note that NN h is an affine mapping Y .˝I S/ ! L .˝I R /. An important modification for S noncompact but locally compact, in particular for S D Rm, combines Theorem C.1.1 for the one-point compactification of S with fine techniques based on relative L1-weak compactness, in particular also Theorem B.3.8, resulting in the following lemma. m Lemma C.1.2 (J.M. Ball [41]). Let uk W ˝ ! R be measurable for every k 2 RN and suppose that the sequence .uk/k2N is tight in the sense that v. . / / < v RC RC supk2N ˝ juk x j dx 1 for some W ! nondecreasing with lima!1 v.a/ D1. Then there exist a subsequence, denoted again by .uk/, and a m m family of probability measures WD f xgx2˝ on R such that for every v 2 C0.R /, Z 1 v.uk/ ! v weakly* in L .˝/; where v .x/ WD v.s/ x.ds/: (C.1.9) Rm

Besides, for every Carathéodory function h W ˝Rm ! R, we have

1 Nh.uk/ ! h weakly in L .˝/ (C.1.10)

1 whenever .Nh.uk//k2N is relatively weakly compact in L .˝/; recall that h is deﬁned in (C.1.5) with S WD Rm.

C.2 Convex local compactiﬁcations of Lp-spaces

The Lebesgue spaces are deﬁnitely the most prominent function spaces occurring in applications. Following [520, 521], we brieﬂy present a fairly universal construction of their locally compact envelopes that are also metrizable and convex in a natural linear space, imitating thus most of the important properties of Euclidean spaces (with the exception that these envelopes are not linear spaces but only convex subsets of those).

3More precisely, this is essentially the concept from [623], while in the original work [622], Young used functions x 7! C.S/, which is already close to the concept of parameterized measures when one identiﬁes C.S/ with M .S/ by Riesz’s theorem. Measure theory itself was not yet launched, however. 618 C Young Measures and Beyond

Considering the Lebesgue space Lp.˝I Rm/, we define a normed linear space ˚ Carp.˝I Rm/ WD h W ˝Rm ! R j h.; s/ measurable; h.x; / continuous; « 9 a 2 L1.˝/; b 2 R Wjh.x; s/jÄa.x/ C bjsjp (C.2.1) of “test Carathéodory integrands” and equip it with the norm h p.˝ Rm/ WD inf a 1.˝/ C bI (C.2.2) Car I jh.x;s/jÄa.x/Cbjsjp L more precisely, as usual, we consider equivalence classes up to zero-measure sets of such integrands. The essential trick is to consider a sufficiently large (but preferably still separable) linear subspace H Carp.˝I Rm/ to define the embedding Z Á p.˝ Rm/ . ; . // ; iH W L I ! H W u 7! h 7! h x u x dx (C.2.3) ˝ and eventually to put

p .˝ Rm/ . p.˝ Rm//: YH I WD the weak* closure of iH L I (C.2.4)

p .˝ Rm/ The elements of YH I are referred to as Young functionals. Proposition C.2.1 (Convex locally compact envelopes of Lp-spaces). The set p .˝ Rm/ YH I is always convex in H . Assuming that H contains at least one coercive . ; / p p .˝ Rm/ integrand, i.e., H 3 h0 with h0 x s jsj , then YH I is locally compact p.˝ Rm/ in H and L I itself is embedded into it (norm,weak*)-continuously via iH deﬁned by (C.2.3). Moreover, if H is rich enough, namely if one of the following situations holds, (a) H fg ˝ vI g 2 C.˝/;N v linear Rm ! Rg[f1 ˝jsjpg;1

Proposition C.2.2 (Extension of Nemytski˘ı mappings). Let p0 >1(respectively m1 m0 p0 D 1). If a W ˝R ! R satisﬁes (B.3.12) with j D 1 and the linear subspace H Carp1 .˝I Rm1 / is sufﬁciently rich such that a g W .x; s/ 7! a.x; s/ g.x/ for

4 More speciﬁcally, iH is (norm,weak*)-homeomorphic. C Young Measures and Beyond 619

0 every g 2 Lp0 .˝I Rm0 / (respectively g 2 C.˝N I Rm0 /), then the Nemytski˘ı mapping p1 m1 p0 m0 Na W L .˝I R / ! L .˝I R / admits an afﬁne continuous extension NN a from p1 .˝ Rm/ p0 .˝ Rm0 / M .˝ Rm0 / YH I to L I (respectively to N I ) deﬁned by Z Z Á ŒNN a g dx WD h; a gi respectively gŒNN a.dx/ WD h; a gi : ˝ ˝N

Remark C.2.3 (Convex compactifications of balls in Lp-spaces). Modifying (C.2.4) p .˝ Rm/ . / as YH;% I WD the weak* closure of iH B% , where B% WD f u 2 p m L .˝I R / jkukÄ% g, we get a convex compactification of the ball B%. p.˝ Rm/ p .˝ Rm/ For two subspaces H2 H1 of Car I , YH1;% I is a finer convex p .˝ Rm/ compactification than YH2;% I , the affine continuous surjection being (the restriction of) the adjoint mapping of the inclusion H2 ! H1. This class of convex compactifications is a lattice, the supremum and the infimum being given respectively by n o p .˝ Rm/; p .˝ Rm/ p .˝ Rm/; sup YH1 I YH2 I D YH1CH2 I (C.2.5a) n o inf Yp .˝I Rm/; Yp .˝I Rm/ D Yp .˝I Rm/; (C.2.5b) H1 H2 HN 1\HN 2

p m where HN jRis the closure of Hj in Car .˝I R / with respect to the seminorm jhj% WD . ; . // supu2B% j ˝ h x u x dxj. Example C.2.4. Following DiPerna and Majda [162], we take G D C.˝/N , V D f v 2 C.S/ j v./=.1 Cjjp/ has a continuous extension on S g, where S is a certain metrizable compactiﬁcation of S Rm.5 Then we consider H D G ˝ V WD spanf g ˝ v j g 2 G;v2 V g. It was essentially proved in [333] that after a certain p .˝ Rm/ M .˝ / rearrangement, 2 YH I has a representation 2 N S of the form Z h.x; s/ ; h . x s/; h iD p d d (C.2.6) ˝N S 1 Cjsj where for s 2 S n S, h.x; /=.1 Cjjp/ is considered extended by continuity and where the measure satisﬁes

Z n ˇ o .; ds/ ˇ 0; 1; L d x ˝N ˇ . x S/ 0 0: p D 2 f g D D (C.2.7) S 1 Cjsj

5If S is bounded, then one can take S aclosureofS in Rm.IfS D Rm, one can consider S the one-point compactification of the locally compact space Rm.Form D 1, the usual two-point compactification S D R [f˙1gof R is finer and still metrizable. We note that the celebrated Stone–Cechˇ compactification ˇRm is not metrizable, however. 620 C Young Measures and Beyond

Let us emphasize that (C.2.7) is a precise characterization of those that are weakly* attainable from Lp.˝I Rm/. Those are called DiPerna–Majda measures. Replacing ˝ with .0; T/˝ and considering suitable normed linear spaces of Carathéodory integrands .I˝/Rm ! R, we can straightforwardly modify the previous construction of the convex locally compact envelopes for Lp.I˝I Rm/ Š Lp.0; TI Lp.˝I Rm// or even for Lq.0; TI Lp.˝I Rm// instead of Lp.˝I Rm/. In applications to rate-independent processes, one needs to compactify B.Œ0; TI Lp.˝I Rm// rather than L1.0; TI Lp.˝I Rm//, however. More speciﬁcally, one needs rather to compactify only bounded sets in B.Œ0; TI Lp.˝I Rm//,say the balls n ˇ o ˇ .Œ0; p.˝ Rm// ˇ N . / : B WD u2B T I L I 8 t2I Wku t kLp Ä (C.2.8)

One simple construction is to consider H as before and the Cartesian product .H/Œ0;T equipped with the Tikhonov product topology, here counting the weak* topology on H. Let us deﬁne n ˇ oÁ ˇ p .˝ Rm/ p.˝ Rm/ ˇ . / : YH; I WD w*-cl iH u2L I ku t kLp Ä (C.2.9)

p .˝ Rm/Œ0;T . /Œ0;T The set YH; I is always convex and compact in H .ThesetB is p .˝ Rm/Œ0;T embedded into YH; I by . . // .Œ0; p.˝ Rm// p .˝ Rm/Œ0;T Œ0;T i W u 7! iH u t t2Œ0;T W B T I L I ! YH; I H (C.2.10)

. / . /Œ0;T with iH from (C.2.3). It can be shown that the closure of i B in H p .˝ Rm/Œ0;T 6 . p .˝ Rm/Œ0;T; ;. /Œ0;T/ is just YH; I . Thus the triple YH; I i H forms a convex compactification of B . This compactification is used in a nontrivial way in Tikhonov’a theorem (Theorem A.2.2) and is, however, not metrizable even if H is separable, because Œ0; T is uncountable. Remark C.2.5 (Finer convex compactifications). Some other, finer convex compactifications of B.Œ0; TI Lp)-spaces have been devised in [530], following the construction of time-correlated Young measures as introduced in [146, Sect.7] (under the name “compatible systems of generalized Young measures”) and further used in [144, 145, 186, 187].

6 p .˝ Rm/Œ0;T . / Every 2 YH; I canR be attained by a net u˘;F;k .˘;F/Œ0;TH ﬁnite;k2N with u˘;F;k 2 B chosen such that h.t/; hi ˝ h.x; u˘;F;k.t; x// dx Ä 1=k for every h 2 F and t 2 ˘. Note that N p .˝ Rm/ even nonmeasurable mappings I ! YH; I can be attained in this way. C Young Measures and Beyond 621

C.3 Suppression of concentration effects

p .˝ Rm/ . / We say that 2 YH I is p-nonconcentrating if there is a sequence uk k2N . / p N such that D w*- limk!1 iH uk and fjukj j k 2 g is weakly relatively compact ı 1 p m in L .˝/. Let us denote the set of all such ’s by YH.˝I R /. ı p m p If H is separable, then every 2 YH.˝I R / has a L -Young measure representation in the sense that there is a weakly*R measurable mapping x 7! x with x a Rm p . / 1.˝/ probability measure on such that x 7! Rm jsj x ds belongs to L and Z Z ˝ ˛ 8h 2 H W ; h D h.x; s/ x.ds/ dxI (C.3.1) ˝ Rm R Œ . / . ; / . / ˝ see [520, Prop. 3.4.15]. We have h x D Rm h x s x ds for a.a. x 2 . p.˝ Rm/ p .˝ Rm/ Every sequence in L I that attains a p-nonconcentrating 2 YH I p .˝ Rm/ does not concentrate energy, provided the convex local compactification YH I in question is fine enough, i.e., if H is large enough. Somewhat more generally, let p m .uk/k2N be a bounded sequence in L .˝I R / such that each weak* cluster point of . . // iH uk k2N in H is p-nonconcentrating and let H be sufficiently rich, namely let H p 0 m contain H0 WD C.˝/N ˝ .R0/ with R0 the smallest complete subring of C .R / p containing constants. Then the set fjukj j k 2 N g is relatively weakly compact in L1.˝/;cf.[520, Prop 3.4.16]. ı ı p m We say that 2 YH.˝I R / is a p-nonconcentrating modification of 2 p .˝ Rm/ ;ı ; . ; / . / YH I if h hiDh hi holds for every h 2 H such that jh x s jÄa x C p 1 C o.jsj / with some a 2 L .˝/ and o W R ! R satisfying limr!1 o.r/=r D 0. Proposition C.3.1 (See [520, Prop. 3.4.17–18, Lem. 4.2.3]). p .˝ Rm/ (i) Every 2 YH I can have at most one p-nonconcentrating modification ı p .˝ Rm/ 2YH I . Moreover, if H is separable, then also: ı p .˝ Rm/ p .˝ Rm/ (ii) The p-nonconcentrating modification 2YH I of 2YH I exists. ı (iii) h ; hi0, provided there is h 2 H such that h.x; s/ a0.x/ for some 1 a0 2 L .˝/. ı ı (iv) h ; hi >0, provided 6D and h 2 H is coercive in the sense that p 1 h.x; s/ a0.x/ C bjsj with some a0 2 L .˝/ and b >0. 1 m 1 m Specific examples arise for H D L .˝I C0.R // or H D L .˝I Cp.R // with m m p p Cp.R / WD f v 2 C.R / j limjsj!1 c.s/=.1Cjsj / D 0 g, yielding L -Young measures defined before (C.3.1), or the DiPerna–Majda measures mentioned above. The following useful assertions are demonstrations of the above theory for simple proofs; cf. also Proposition 4.1.5 on p. 248 for a more advanced use of this technique. Proposition C.3.2. Let v W Rm ! R be continuous and 0 Ä v.s/ Ä C.1 Cjsjp/, p m 1 m p >1, andR let .uk/k2N be boundedR in L .˝I R / and uk ! uinL .˝I R /. Then lim infk!1 ˝ v.uk/ dx ˝ v.u/ dx. 622 C Young Measures and Beyond

1 m Proof. Consider H WD span..C.˝/N ˝ V/ [ L .˝I Cp.R /// with a suitable 7 p .˝ Rm/ . / V, and a weak* limit 2 YH I of a subsequence of uk k2N. Then consider the DiPerna–Majda representation from Example C.2.4, here extended 1 m m on L .˝I C0.R //. Let us decompose D 0 C 1 with supp. 0/ ˝N R and m m supp. 1/ ˝N .R nR /. Taking a test function h.x; s/ WD jsu.x/j and taking into account that the continuous extension of h.x; /=.1Cjjp/ on Rm is zero, since 1 m p >1, and that uk ! u converges in L .˝I R /, we get Z 0 . ; . // uk u L1.˝IRm/ D h x uk x dx ˝ Z Z h.x; s/ h.x; s/ . x s/ 0. x s/: ! p d d D p d d (C.3.2) ˝N Rm 1 Cjsj ˝N Rm 1 Cjsj

p Since the last integral must vanish and 0 0 and h.x; /=.1 Cjj / 0 vanishes only at u.x/, we can see that 0.x; / is supported at u.x/ for a.a. x 2 ˝. Then, since v 0 is assumed and since always 1 0,wehave Z Z v.s/ v.u / x . x s/ lim k d D p d d k!1 ˝ ˝N Rm 1 Cjsj Z Z v.s/ v.s/ 0. x s/ 1. x s/ D p d d C p d d ˝Rm 1 Cjsj ˝N . RmnRm/ 1 Cjsj Z Z Z v.s/ v.u.x// x 1. x s/ v.u.x// x: D d C p d d d ˝ ˝N . RmnRm/ 1Cjsj ˝ (C.3.3)

By a contradiction argument, one can show that in fact, (C.3.3) holds not only for the subsequence selected above but for the whole sequence .uk/k2N. ut The next assertion is about suppressing oscillations of minimizing sequences of strictly convex functionals, which was ﬁrst realized by A. Visintin [607], cf. also [609, Sect. X], although the proof used a different technique for a more general situation.8 Here, using Young measure, one can prove the following result. Proposition C.3.3 (Strong convergence of minimizing sequences). Let p m 1 m .uk/k2N be a sequence in L .˝I R / such that uk * uinL .˝I R / and

7Without going into details, we can assume that V contains v and is such that f v. /=.1 Cj jp/ j v 2V g is a complete separable ring of bounded continuous functions containing 1. Then Rm v. /=.1 p/ Rm corresponding to this ring is metrizable, and Cj j can be continuouslyR extended on . 8 1;p Analogous arguments can be used also for the functional u 7! ˝ h.x; ru.x// dx on W .˝/, p >1, where in addition, the constraint id Dru is to be considered; cf. [520, Chap. 5]. In [324, Theorem 3.10], a direct construction of a modiﬁed sequence having a relatively L1-weakly compact energy of gradients is performed using a Hodge decomposition. See also [191]fora general investigation of oscillations and concentration in the vectorial case. C Young Measures and Beyond 623 R R p m limk!1 ˝ h.x; uk.x// dx D ˝ h.x; u.x// dxforsomeh2 Car .˝I R / satisfying p h.x; s/ jsj ,p>1, and h.x; / strictly convex for a.a. x 2 ˝. Then uk ! uin Lp.˝I Rm/. Sketch of the proof. We can consider a separable linear space H containing h and 1 m also L .˝I C0.R //. Since h is coercive, the minimizing sequence .uk/k2N induces, p .˝ Rm/ ; when embedded into H , a Young functional 2YH I that minimizes h hi on p .˝ Rm/ YH I and is p-nonconcentrating due to Proposition C.3.1(iv). Otherwise, its ı p .˝ Rm/ p-nonconcentrating modiﬁcation 2YH I would yield a strictly smaller value of h; hi, which would contradict the minimality of . Then has a Young-measure representation in the sense of (C.3.1). The strict convexity of h.x; / implies that is composed from Dirac measures, i.e., x D ıu.x/ for a.a. x 2 ˝ with some u 2 Lp.˝I Rm/. Otherwise would again not be a mimimizer of h; hi. Then we use p a test integrand hu.x; s/ WD js u.x/j , and from hi.uk/; hui!h ; hui we read that Z Z ˇ ˇ ˝ ˛ ˝ ˛ ˇ ˇ ˇ ˇp ˇ ˇp uk.x/u.x/ dx D i.uk/; hu ! ; hu D u.x/u.x/ dx D 0: ˝ ˝ ut References

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Symbols assumptions of general use -convergence, 90 (AC1)-(AC2): approxim. compatibility, 87 simultaneous, 356 (C1)-(C2): compatibility, 55 ˇ-differentiable, 357 (D1)-(D2): about dissipation, 46 -algebra, 591 (E1)-(E2): about stored energy, 47 -ﬁnite complete measure space, 592 (GC1)-(GC2): generalized compatibility, 83 (I1)-(I2): about reduced functionals, 75 (RC1)-(RC2): reduced compatibility, 77 A attainable point, 581 A-quasiconvexity, 247 Aubin–Lions lemma, 609 a.e.-local solution, 36, 132, 230 austenite, 285 to problems with viscosity/inertia, 462 available driving force, 17 absolutely continuous, 595 absolutely continuous mapping, 603 actual driving force, 17 B adhesive contact, 297 back stress, 313 dynamical, 513 backtracking, 206 small strains, 395 balanced-viscosity solution, 36, 222 with friction, 523 Banach algebra, 535, 584 adjoint operator, 586 Gelfand representation, 536 admissible, 7 ordered, 589 aging, 502 unital, 535, 584 algebra, 591 Banach selection principle, 586 almost all (a.a.), 594 Banach space, 584 almost everywhere (a.e.), 594 ordered, 588 alternating minimization algorithm, 209 bang-bang-type delamination, 292, 296 Ambrosio–Tortorelli functional, 368, 525 base of a topology, 580 anisotropy energy, 439 binomial formula, 162, 199, 483 approximable solution, 36, 222 Biot equation, 3, 18 approximate incremental problem, 86 Bochner integrable, 604 convergence, 88 Bochner measurable, 603 strengthened, 86 Bochner space, 606 approximate minimizers, 85 Bolzano–Weierstrass theorem, 582 approximate-stability set, 86, 107 Borel -algebra, 592 approximately stable sequence, 87 Borel measure, 592

© Springer Science+Business Media New York 2015 651 A. Mielke, T. Roubícek,ˇ Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences 193, DOI 10.1007/978-1-4939-2706-7 652 Index boundary, 580, 600 continuous, 181 bounded deformations, 316 epigraph, 90 bounded set, 584 in measure, 596 bounded variation, 306, 603 Moore–Smith, 57, 581 BV solution, 36, 222, 226 Mosco, 181 two-scale, 343 variational, 90 C convex compactification, 583 calibration, 159 of balls in Lp-spaces, 619 Carathéodory mapping, 598 convex functional, 587 Cauchy net, 584 partial/separate, 151 Cauchy–Green stretch tensor, 237 proper, 587 Cauchy–Schwarz inequality, 586 convex locally compact envelope, 618 causal, 7 convex set, 587 CD solution, 35, 131 convex subdifferential, 12, 129, 588 chemical potential, 529 Cosserat continuum, 331 Ciarlet–Necasˇ condition, 242, 264 Cosserat couple modulus, 331 for delamination, 293 Coulomb friction, vii, 429, 518 in crack problem, 306 crack, 26 Clapeyron principle, 397 prescribed-path, 290 Clarke subdifferential, 129 crack-transfer lemma, 308 Clausius–Duhem inequality, 565, 576 cross-quasiconvex, 247 abstract, 559 cross-quasiconvexification, 389 closed, 580 closure, 580 cluster point, 581 D coarser topology, 581 damage, 309, 346 coercive, 117 by swelling, 529 force, 441 cohesive, 368 cofactors, 239 complete, 356 cohesive contact, 302 ductile, 379 coldness, 559 in Jeffreys materials, 510 compact, 581 in plasticizable materials, 379 locally, 581 in viscoelastic materials, 505 sequentially, 581 regularity, 351, 508 compactification, 583 stress-driven, 509 coarser, 583 with healing, 500, 532 convex, 583 Darboux sum, 605 finer, 583 Darcy law, 528 compatibility conditions, 55 debonding, 290 complete damage, 356 defect measure, 133, 229 generalized energetic solution, 365 deformation, 237 composite material, 104 bounded, 316 concatenation property, 7, 49 deformation gradient, 237, 238 conditioned continuity delamination, 290 of the power of the external forces, 55 at small strains, 394, 512 cone, 587 bang-bang type, 292 conjugate exponent, 596 brittle, 294, 395, 425, 516 contact set, 224 BV-bound on u, 427 continuous mapping, 582 local solution, 404 minors, 245 mixed-mode sensitive, 400, 406, 428 continuous operator, 585 engineering model, 403 convergence mixity-sensitive -, 90 engineering model, 515, 577 Index 653

regularity, 426 duality pairing, 585 weakening, 303 Dunford–Pettis compactness criterion, 598 with healing, 534 with inertial/visco-effects, 512 demagnetizing field, 439 E dense, 581 earthquakes, 504 depolarizing field, 456 elastic modulus, 239 deviatoric part of a tensor, 313 of adhesive, 297, 399 diameter, 594 elasticity domain, 15 differential solution, 35, 131, 146 elastoplasticity, 21, 313 to friction problem, 434 dynamic, 497 Dini subdifferential, 129 finite-strain, 250 DiPerna–Majda measures, 620 energetic rate-independent system, viii, 9, Dirac measure, 592 45 direct method in the calculus of variations, 51 energetic solution, 36, 48, 546 directed set, 579 absolute continuity, 178 direction of easy magnetization, 440 continuity, 144 directional derivative, 81, 587 continuous dependence on data, 169 Dirichlet boundary conditions, 241 generalized, 82 displacement, 237 regularity, 484 Kirchhoff–Love, 337 symmetry-breaking, 375 dissipated energy to damage at large strains, 311 effective, 401, 507 to delamination model, 298, 395 dissipation to fracture model, 369 hardening, 212 to plasticity at large strains, 254 total, 47 to Prandtl–Reuss plasticity, 317 weakening, 211 to problems with viscosity/inertia, 463 dissipation distance, viii, 46, 75 existence, 464, 492 dissipation function, 123 to shape-memory-alloy model, 268 dissipation metric, 120, 123 to thermodynamical system, 546 in problems with viscosity, 490 uniqueness, 146, 150, 484, 488 dissipation potential, 2, 12, 123 energetic solutions, viii, 24 p-homogeneous, 12 energy balance (E), 16, 48, 76 translation-invariant, 120 for Kelvin–Voigt material, 240 dissipation pseudopotential, 2, 15 total for thermodynamic system, 543 dissipation rate, 15, 123 weakened, 82 dissipative component, 13 energy-dissipation principle, 19 distance, 581 energy-storage functional, viii intrinsic, 124 wiggly, 28 path, 124 enthalpy transformation, 542 distortion matrices, 287 entropy, 540 distributional derivative, 599 equation, 540, 559, 565 distributional time derivative, 608 epigraph, 587 divergence, 601 convergence, 90 domain, 599 essential infimum, 595 Lipschitz, 600 essential supremum, 595 of Ck-class, 600 Euclidean space, 584 of a functional, 47, 588 Euler–Lagrange equation, 259 doubly nonlinear equation, 42 evolutionary system, 7 driving force, 17 evolutionary -convergence of ERIS, 100 dual ordering, 588 evolutionary variational inequality, 14 dual space, 585 evolutionary variational inequality, 169 to Bochner space Lp.0; TI V /, 607 exchange energy, 439 654 Index

F global stability (S), 48 Fan Glicksberg theorem, 586 gradient, 599 fatigue, 43, 393 gradient flow Fatou theorem, 597 generalized, 18 Fenchel–Young inequality, 588 viscous, 18, 33 ferroic, 438 gradient system multi-, 453 generalized, 17 Filippov selection, 593 viscous, 17 fine structure, 278, 444 gradient theory, 267 fineness of a partition, 51 in damage, 310 finer convex compactification, 583 in plasticity, 250, 329 finer net, 580 graph of a (set-valued) mapping, 592 finer topology, 581 Green formula, 602 finite interpenetration, 430, 518 on a surface, 602 flow rule, 18 Green–Lagrange strain, 239, 287 Fourier law, 538, 541 Griffith criterion, 292 Fréchet differential, 587 group of orientation-preserving rotations, 238 Fréchet subdifferential, 129 fractional-step strategy, 207 for plasticity with damage, 501 H for thermodynamical systems, 548 Hamilton’s variational principle, 498 for visco/inertial systems, 465 hardening, 212, 250, 254, 255, 313, 496 fracture, 368 in ferromagnets, 449 in viscoelestic material, 525 hardening parameters, 245 frame indifference, 250 Hausdorff, 584 frame-indifference, 238, 249 Hausdorff measure, 594 for dissipative forces, 241 healing, 349, 415, 500, 531 for gradient theory, 271 heat capacity, 541 for nonlocal terms, 277 heat-transfer equation, 541 Fréchet differential, 130 Heisenberg constraint, 439 friction, 20, 429, 517 Helly selection principle, 72, 610 dynamic vs static, 524 generalized, 72 on adhesive contact, 523 Helmholtz decomposition, 455 functions of bounded variation, 306 Hilbert space, 584 special, 306 holder-inequality, 596 homogenization, 104 counterexample, 190 G for linearized elastoplasticity, 342 Gâteaux differential, 587 with quadratic energies, 189 gamma convergence, 90, 91 hyperbolic, 486, 497 strong, 91 hyperelastic materials, 238 weak, 91 hyperstress, 267 Gâteaux differential, 130 hysteresis operator, 7 Gelfand representation, 536, 585 identification, 176 Gelfand triple, 544, 608 Krasnoselski˘ı–Pokrovski˘ı, 176 general linear group, 242 Prandtl–Ishlinski˘ı, 172 generalized standard materials, 236 Preisach, 175 generalized energetic solution, 82 for complete damage, 365 generalized gradient flow, 18, 527 I generalized gradient system, 17 identification generalized Prandtl–Ishlinski˘ı operator, 173 of hysteresis operators, 176 Gibbs simplex, 287 of rate-independent systems, 159 Gilbert equation, 443 improved stability estimate, 145 Index 655 incremental problem, 26, 51, 191 laminate, 282, 285 approximate, 86 Landau–Lifshitz equation, 443 decoupled, 467, 548 Laplace–Beltrami operator, 602 finite-dimensional, 191 Lavrentiev phenomenon, 264 strengthened approximate, 86 Lebesgue measurable, 594 indicator function ıK , 589 Lebesgue measure, 594 induced distance, 123 Lebesgue outer measure, 593 induced metric, 124 Lebesgue point, 603 infimum, 580 Lebesgue space, 595 inner product, 584 uniformly convex, 596 input–output system, 7 Lebesgue theorem, 597 integrable function, 594 left derivative, 76 integral, 594 Legendre–Fenchel transform, 588 integration by parts formula, 608 limit, 581 interaction energy, 439 inferior or superior, 582 interface plasticity, 400 simultaneous -, 356 interfacial energy, 271 slow-loading, 488 nonlocal, 276 limiting subdifferential, 129 interior, 580 linear operator, 585 internal energy, 540, 544 linear ordering, 579 internal parameters, 236 linearized elastoplasticity, 21 internal variable, 14, 236, 249 Lipschitz domain, 600 internal variables, 244 local stability, 16 interpolant local solution, 36, 132 piecewise affine, 156, 466 a.e.-, 132, 230 piecewise constant, 53, 153, 192 force-driven, 374 intrinsic distance, 124 maximally dissipative, 138, 222, 372 irreversibility, 304 counterexample, 376 irreversible crack evolution, 304 symmetry breaking, 376 irreversible quasistatic evolution, 46 to brittle delamination, 398 isotropic hardening, 255, 313, 449 to delamination, 405 isotropic material, 240, 243, 561 to delamination in mixed modes, 408 to plasticity with damage, 382 local stability, 16, 24 J local stability condition, 132 jump-transfer lemma, 305, 308 locally convex space, 584 complete, 584 lower semicontinuity, 582 Löwner ordering, 589 K Kelvin–Voigt material, 240 vanishing viscosity, 421 M kinematic hardening, 254, 313 magnetic constant, 439 kinetic energy, 461 magnetostatic energy, 439 Kirchhoff stress, 253, 257, 274 magnetostrictive materials, 453 Kirchhoff–Love displacement, 337 mapping Korn inequality, 244 nondecreasing, 579 Kuratowski–Zorn lemma, 580 nonincreasing, 579 set-valued, 592 martensite, 285 L maximal, 579 Lamé coefficients, 240 maximal monotone operator, 120 Lamé system, 243 maximal responsive mapping, 122 656 Index maximally dissipative local solution, 138, 234 mutual uniform ˛-convexity, 146 alternative definition, 139, 376 mutual-convexity condition, 23, 31 example, 141 to damage, 349 maximum-dissipation principle, 17, 137, 374 N approximate, 158, 379, 383, 410 neighborhood, 580 counterexample, 376 Nemytski˘ı mapping, 598 Maxwell system extension, 618 rest (magnetostatics), 439 on Bochner spaces, 607 measurable function, 594 net, 580 measurable mapping, 592 Neumann boundary conditions, 241 measurable set, 592 non-self-penetration condition, 242 measurable set-valued map, 592 nonassociative, 211 measure, 591 nonconvex elastic energy, 504 absolutely continuous, 595 nondissipative component, 13 Borel, 592 nonsimple material, 267, 332, 427 complete, 592 norm, 583 Dirac, 592 normal cone, 587 Hausdorff, 594 normal integrand, 598 Lebesgue, 594 ŒŒ normal jump n , 394 outer, 592 normal-compliance contact, 429, 437, 518 regular, 591 normed linear space, 584 Young, 113, 278, 616 numerical strategies metric, 581 alternating-minimization, 209 metrizable, 581 backtracking, 206 minimal, 579 for delamination at large strain, 297 minimization dissipation potential, 16, 139 for plasticity at large strain, 261 minimization-energy principle, 238 fractional-steps, 207, 465, 501, 548 minor, 245 incremental minimization, 26, 51, 191 minor hysteretic loop, 448 quadratic nonsmooth terms, 208 mixity of modes, 400 semi-implicit formula, 152, 407 angle of, 403 for plasticity with damage, 380 mixture function, 390 mode-mixity angle, 403 modulus of continuity, 64 momentum equilibrium, 238 O monotone operator, 120 Ogden-type material, 239 maximal, 120 example, 254 Moore–Smith convergence, 581 one-sided limits, 59 Moore-Pollard-Stieltjes integral, 606 open, 580 Mosco convergence, 91, 181 operator Mosco transformation, 206 adjoint, 586 multiferroic, 453 hysteresis, 172, 175 multiplicative decomposition, 251 Krasnoselski˘ı–Pokrovski˘ı, 176 multiplicative split, 242 linear, 585 multiplicative stress control, 253 memory, 9 mutual recovery sequences, 308 play, 20 mutual recovery sequence, 62, 97 Prandtl–Ishlinski˘ı, 172 for damage, 312, 507 Preisach, 175 for delamination problem, 295, 514 stop, 171 numerical approximation, 301 stop vector-valued, 171 for plasticity at large strains, 259 trace, 601 for vanishing-hardening plasticity, 321 vector-valued play, 21 for viscous problems, 153 optimal jump paths, 225 Index 657 ordering, 579 premonotone mapping, 122 dual, 588 principle Löwner, 589 Hamilton’s, 498 of compactifications, 583 of energy minimization, 238 of convex compactifications, 583 of maximum dissipation, 17, 137, 374 of topologies, 581 of minimum dissipation potential, 16, 139 on a Banach algebra, 589 probability measure, 592 outer measure, 592 projective topology, 582 projector, 585 proper functional, 587 P pseudoelasticity, 285 p-homogeneous, 12 pseudopotential of dissipative forces, 240 dissipation potential, 120 pseudopotential of viscous forces, 461 p-nonconcentrating modification, 621 parameterized solution, 36 parent hysteretic loop, 448 Q path distance, 124 quadratic mathematical-programming, 208 penalty function, 92, 276 quadratic trick, 63, 160, 162 Pettis theorem, 604 in micromagnetism, 572 phase transformation, 285 quasiconvex, 238 phase-field variable, 267, 530, 569 A-, 247 piecewise constant interpolants, 53, 192 cross-, 247 piezoelectric materials, 456 lower semicontinuity, 248 Piola–Kirchhoff stress, 238 quasidistance, 581 plastic indifference, 250 quasiplasticity, 285 plastic strain, 245 quasistatic crack evolution, 304, 308 plastic tensor, 250 quasistatic evolution, 460 plasticity irreversible, 46 at large strains, 21, 250 quasivariational inequality, 133, 519 by swelling, 526 gradient, 250, 330, 379 interface, 400 linearized, 313, 496 R multiple thresholds, 331 Radon measure, 591 perfect, 316, 568 Radon–Nikodým theorem, 595 with damage, 502 rank-1 condition, 282 stress-driven, 498, 567 rate-independent, 7 with aging, 502 rate-independent system, ix, 9, 45 with damage, 379, 499, 534 arising by slow-loading limit, 488 with hardening, 250, 313, 496 arising by vanishing viscosity, 38, 215 play operator, 20, 171 calibration of, 159 vector-valued, 21 energetic, viii, 9, 45 Poisson ratio, 243 identification of, 159 polyconvex, 238 reduced, 13 lower semicontinuity, 246 rebonding, 534 positive measure, 592 recovery sequence, 91 positive variation, 592 reduced compatibility condition, 77 positivity of temperature, 557 reduced energy functional, 68, 75 power balance, 16 reduced functional, 45, 54 Prandtl–Ishlinski˘ı operator, 172 reduced power, 54 Prandtl–Reuss (perfect) plasticity, 316 reduced problem, 13 thermodynamics, 568 reduced rate-independent system, 68 predual, 585 reduced RIS, 13 Preisach operator, 175 reduced stability sets, 68 658 Index reference configuration, 237 magnetic, 453 reflexive space, 586 polycrystals, 387 regular measure, 591 simple function, 594 regularity simple mapping, 603 in z-variable, 484 simultaneous -limit, 356 in small-strain plasticity, 316 singular perturbation, 112, 278, 444 inviscid hyperbolic case, 486 slide, 140 of damage, 351, 508 slide solution, 37, 140 of delamination, 426 slow-loading limit, 488 of displacements, 507 small-strain tensor, 243 of energetic solutions, 162, 484 smart material, 284 relaxation, 107 Sobolev exponent, 600 by lower semicontinuous envelope, 110 Sobolev space, 599 by Young measures, 113, 279, 444 Sobolev trace exponent, 601 relaxed problem, 114, 281, 445 Sobolev–Bochner space, 608 relay operator, 174 Sobolev–Slobodecki˘ı space, 602 Rellich–Kondrachov theorem, 600 solution remanent polarization, 456 a.e.-local, 36, 132, 230, 462 responsive set-valued operator, 122 maximally dissipative, 222, 234 restriction property, 7, 49 approximable, 36, 222 Riemann sum, 604 balanced-viscosity (= BV), 36, 222 Riemann–Stieltjes integral, 604 BV, 36, 222, 226 Riesz theorem, 592 CD , 35, 131 RIS (= rate-independent system), 9 differential, 35, 131, 146 Robin boundary conditions, 241 energetic, 36, 48, 546 rotation of a vector field, 439 generalized, 82 local, 36, 132 maximally dissipative, 138, 372 S parameterized, 36 C (S )-property, 152, 480 semi-energetic, 133, 229 integral variant, 464 semidifferential, 131, 146 of p-Laplacian, 158 slide, 37, 140 safe-load condition, 177 V-approximable, 215 saturation magnetization, 439 semi-energetic, 229 scalar play operator, 171 V-parameterized, 216 scalar product, 584 vanishing-viscosity, 36, 213 second-order cone programming, 208 weak, 37, 132 selection of a set-valued map, 592 St. Venant–Kirchhoff material, 287 self-adjoint linear functional, 589 St. Venant–Kirchhoff material, 239 self-controlling models, 48 stability semi-energetic solution, 133, 229 global, 48 semi-implicit formula, 152, 465, 491 local, 16, 24, 132, 140 semidifferential solution, 131, 146 semi-, 139 seminorm, 584 stability set, 49 semistability, 139, 463 stability set in problems with generalized viscosity, 490 closedness of, 55 semistable sequence, 154, 491 stable sequence, 54, 77, 95 separable, 581 approximately, 87 sequence, 580 for a sequence of problems, 95 sequential laminate, 283 semi-, 154 sequential semicontinuity, 582 stable states, 49 sequentially continuous mapping, 582 state space, 45 set-valued map, 592 stop operator, 171 shape-memory alloys, 286 vector-valued, 171 Index 659 stored energy, 238 uniformly integrable set, 597 multiwell, 288 unilateral contact, 291, 512 stress at large strains, 265 back, 313 with friction, 429 hyper-, 267 uniqueness Kirchhoff, 253 counterexample, 375 Piola–Kirchhoff, 238 of energetic solution, 146, 150 strict convergence, 318 of solutions to linearized plasticity, 314 strictly convex functional, 587 of viscous delamination, 420 stronger topology, 581 unit outward normal, 601 subdifferential upper semicontinuity, 582 Clarke, 129 convex, 12, 129, 588 Dini, 129 V Fréchet, 129 V-approximable solution, 215 limiting, 129 semi-energetic, 229 sublevel set, 48 V-parameterized solution, 216 sum rule, 588 vacuum permeability, 439 superconductivity, 452 vanishing viscosity supremum, 580 approach, 213 surface divergence divS, 602 contact potential, 223, 224 surface gradient rS, 602 in dynamical systems, 490 sweeping process, 3, 21 in Kelvin–Voigt material, 421 swelling, 526 RIS, 38, 215 stress-driven, 530 vanishing-viscosity limit, 20 solution, 36 T variable tangent cone, 587 (non)dissipative, 13 ŒŒ tangential jump t , 394 intensive vs. extensive, 544 Tikhonov product topology, 582 internal, 14, 236 Tikhonov theorem, 583 variation, 223, 603 topological space, 580 of a measure, 591 topology, 580 special functions of bounded, 306 compact, 581 variational convergence, 90 Hausdorff, 580 variational inequality, 14 locally compact, 581 viscous gradient flow, 18, 33 relative, 580 viscous gradient system, 17 relatively compact, 581 Vitali theorem, 597 sequentially compact, 581 volume fraction, 444 strong, 584 weak, 586 total dissipation, 47 W in problems with viscosity, 491 weak lower semicontinuity, 586 total variation, 537, 591 weak solution, 37, 132 trace, 601 to damage with healing, 532 translation-invariant D, 124 to micromagnetic problem, 443 translation-invariant R, 120 weak topology, 586 weak* topology, 586 weakened energy balance, 82 U weakening, 211 unidirectional processes, 210, 560 delamination, 303 uniform ˇ-differentiability, 357 weaker topology, 581 uniformly convex space, 584 weakly lower semicontinuous, 117 660 Index weakly measurable, 603 Young functional, 618 weakly* measurable, 603 p-nonconcentrating, 621 wetting/dewetting, 50 Young-measure representation, 621 wiggly energy, 28 Young inequality, 595 Young measure, 113, 279, 444, 616 gradient, 278 Y representation, 621 Yosida–Moreau regularization, 247 Young modulus, 243