Weak convergence of the sequences of homogeneous Young measures associated with a class of oscillating functions
aa r X i v : . [ m a t h . F A ] M a r Weak convergence of the sequences ofhomogeneous Young measures associated with aclass of oscillating functions
Piotr Pucha la
Institute of Mathematics, Czestochowa University of Technology, al. Armii Krajowej 21,42-200 Cz¸estochowa, PolandEmail: [email protected], [email protected]
Abstract
We take under consideration Young measures with densities. The no-tion of density of a Young measure is introduced and illustrated withexamples. It is proved that the density of a Young measure is weaklysequentially closed set. In the case when density of a Young measure isa singleton (up to the set of null measure), it is shown that the strongclosedness (in rca( K )) of the set of such measures, associated with Borelfunctions with values in the compact set K ⊂ R l , is equivalent with thestrong closedness (in L ( K )) of the set of their densities, provided theset K is convex. For an m-oscillating function the notion of a total slopeis proposed. It turns out, that if the total slopes of the elements of thesequence of oscillating functions form monotonic sequence, then the se-quence of the respective (homogeneous) Young measures is weakly con-vergent in rca( K ). The limit is a homogeneous Young measure with thedensity being the weak L sequential limit of the densities of the under-lying Young measures. Keywords : Young measures, weak convergence, density, total slope
AMS Subject Classification : 46N10; 46N30; 49M30; 60A10
Minimization of functionals with nonconvex integrands is an important problemboth from theoretical and practical, including engineering applications, pointsof view. The fact that in this case the considered functional, although boundedfrom below, usually does not attain its infimum, is the source of the maindifficulty when dealing with this kind of task. It is described in the followingexample attributed to Oscar Bolza and Laurence Chisholm Young.
Example 1.1. (see [12]) Find the minimum of the functional J ( u ) = Z (cid:2) u + (cid:0) ( dudx ) − (cid:1) (cid:3) dx, onvergence of the sequences of Young measures with boundary conditions u (0) = 0 = u (1) .It can be shown that inf J = 0 .Consider the function u ( x ) = x, for x ∈ (cid:2) , (cid:1) − x, for x ∈ (cid:2) , (cid:1) x − , for x ∈ (cid:2) , (cid:1) . Then the sequence u n ( x ) := n u ( nx ) is the minimizing sequence for J , that isit satisfies the condition J ( u n ) → inf J . However, we have inf J 6 = J (lim u n ) because if the limit u of ( u n ) were the function satisfying the equality inf J = J ( u ) ,it would have to satisfy simultaneously two mutually contradictory conditions: u ≡ and du dx = ± a.e. (with respect to the Lebesgue measure on [0 , ). Thismeans that J does not attain its infimum. Observe, that the elements of the minimizing sequence are wildly oscillatingfunctions. We will call the functions of this type the m-oscillating functions, seeDefinition 3.1. This is typical situation when one minimizes integral functionalswith non-(quasi)convex integrands.Further motivating examples can be found, among others, in [13] and [19].Generally, the problem can be dealt with in two ways. The first one is con-vexification the original functional (more precisely: quasiconvexification). Thisprocedure saves the infimum of the functional, but it has two main drawbacks.Namely, computing explicit form of the (quasi)convex envelope is usually verydifficult in practice. Further, it erases some important information concerningthe behaviour of the minimizing sequences: calculating a weak ∗ accumulationpoint of the minimizing sequence is calculation a limit of the sequence withintegral elements. The integrands are compositions of the Carath´eodory func-tion with highly oscillating elements of the minimizing sequence. Thus, in somesense, only the mean values are taken into account, see for example [16] andreferences cited there.Another way is connected with the discovery of Laurence Chisholm Young,first published in [20]. He observed that the weak ∗ limit of a sequence withelements being composition of a continuous function with oscillating functions isin fact a set function . This set function can be looked at as a mean summarizingthe spatial oscillatory properties of minimizing sequences. Thus not all theinformation concerning behavioral characteristics of the phase involved is lostwhile passing to the limit.More precisely, let there be given: R d ⊃ Ω – nonempty, bounded open set ofthe Lebesgue measure M > K ⊂ R l – compact; ( u n ) – a sequence of functionsfrom Ω to K , convergent to some function u : weakly ∗ in L ∞ or weakly in L p , p ≥ ϕ – an arbitrary continuous real valued function on R l .Then the continuity of ϕ yields the norm boundedness of the sequence( ϕ ( u n )) of compositions. By the Banach-Alaoglu theorem we infer the existenceof the converging to some function g the subsequence of ( ϕ ( u n )). However, ingeneral not only g = ϕ (( u )), but g is not even a function with domain in R l . onvergence of the sequences of Young measures ν x ) x ∈ Ω Borel probability measures defined on Borel σ -algebra of subsetsof R l , with supports contained in K , satisfying for each continuous ϕ and anyintegrable function w the conditionlim n →∞ Z Ω ϕ ( u n ( x )) w ( x ) dx = Z Ω Z K ϕ ( s ) ν x ( ds ) w ( x ) dx := Z Ω ϕ ( x ) w ( x ) dx, where ϕ ( x ) := Z R l ϕ ( λ ) ν x ( dλ ) . Thus the family ( ν x ) x ∈ Ω can be regarded as the ’generalized weak ∗ limit’ of thesequence ( ϕ ( u n )). Young himself called them ’generalized curves’; today we callthem Young measures (or relaxed controls in control theory) associated with thesequence ( u n ).In the very important special case, where for any x ∈ Ω there holds ν x = ν (that is the family ( ν x ) x ∈ Ω is a one-element only), we use the term homogeneousYoung measure .We can thus look at Young measure as at the generalized limit of thesequence whose elements are compositions of the fixed continuous functionwith the elements of weakly (or weakly ∗ ) convergent sequence of oscillatingfunctions. This approach is exposed in detail in [13], where the existencetheorem (Theorem 6.2) with very general assumptions is stated and proved;one may also consult [12].We can also look at the Young measures from another point of view. Namely,they can be regarded as linear functionals acting on L ( Ω, C ( K )) – the spaceof vector-valued, Bochner integrable functions defined on Ω and taking valuesin the Banach space of continuous real-valued functions defined on a compactset K . This enables us to associate Young measure with any Borel function f : Ω → K , see Theorem 3.1.6 in [19]. Using this approach one may study Youngmeasures in general, with Young measures associated with measurable functionsbeing specific examples. We refer the reader to [19] for detailed exposition ofthis point of view.Although calculating an explicit form of the Young measure is usually verydifficult, it turns out that quite often it can be done relatively easily, whenfunctions under consideration are bounded oscillating ones. This is possibleif the latter of the approaches mentioned above is adopted. As it has beenobserved in [14] and [9], for quite large class of oscillating functions Youngmeasures associated with them are homogeneous, absolutely continuous withrespect to the Lebesgue measure with densities being merely the sums of theabsolute values of the Jacobians of the inverses of their invertible parts. Thesesums (called later the total slopes , see Definition 3.2) are important: different functions with equal total slopes have the same Young measure. Thus in manycases (for example for the sequence f n ( t ) = sin(2 nπt ) , t ∈ (0 , , n ∈ N , akind of ’canonical’ example of the sequence of oscillating functions generating onvergence of the sequences of Young measures f . In [15] these observations has been broadened. Namely, it has been provedthat the weak convergence of the sequence of homogoneous Young measures(understood as elements of the Banach space rca( K )) with densities is equivalentto the L convergence of the sequence of these densities.It is worth mentioning that the above results have been obtained with reallysimple, in comparison to the usual Young measure methods, apparatus: thechange of variable theorem for multiple integrals plays central role. This can beimportant in applications.After the above short presentation of the classical problem of minimizing anintegral functional with nonconvex integrand, whose solution leads directly tothe Young measures and two approaches to the Young measures, we give outlineof the theory needed in the sequel. The third section deals with the sequencesof m-oscillating functions, i.e. functions being sums of diffeomorphisms thathave continuously differentiable inverses. We introduce the notion of a totalslope of such oscillating function and observe that the set of the Young mea-sures associated with the elements of the set m-oscillating functions, definedon an open set Ω ⊂ R d with values in the compact set K ⊂ R l , is relativelyweakly compact. The last section begins with recollection of the basic factsconcerning weak sequential convergence of functions and measures, in particu-lar the theorem of Jean Dieudonn´e. Then we introduce the notion of a densityof a (not necessarily homogeneous) Young measure. We illustrate this notionwith several examples; in particular, one of them makes use of the Bressan-Colombo-Fryszkowski theorem about existence of a continuous selection of amultifunction with decomposable values. We prove that density of a Youngmeasure is a weakly sequentially compact set.The final part of the fourth section is devoted to homogeneous Young mea-sures and again to m-oscillating functions. First it is shown, that if we additio-nally assume convexity of the compact set K and consider such Borel functionsfrom Ω to K , that the Young measures associated with them are homogeneousand absolutely continuous with respect to the Lebesgue measure on K , thenthe norm closedness (in rca( K ) with the total variation norm) of the set ofsuch Young measures is equivalent with the norm closedness (in L ( K )) of theset of their densities. It then follows (already without convexity assumptions on K ), that the sequence of homogeneous Young measures with densities convergesweakly to the measure of the same type if and only if the sequence of the re-spective densities converges weakly to the density of the limit measure. Finally,given a sequence of m-oscillating functions we show, that if their total slopesform monotonic sequence, than the sequence of the associated Young measures isweakly convergent to a homogeneous Young measure. Moreover, this limit mea-sure has a density which is a weak limit of the sequence of densities of the respec-tive Young measures. Thus majority of the examples of the homogeneous Youngmeasures generated by the sequences of oscillating functions, that can be metin the literature, turn out to be illustrations of the special case of that result. onvergence of the sequences of Young measures In this section we gather information about Young measures that will be of usein the sequel. The literature concerning Young measures is large. An interestedreader may consult, for example, besides the books that have already been cited,[1, 2, 6, 8] and the references cited there.In this article we use the approach to Young measures described in [19] inChapter 3.We will denote by rca( K ) the space of regular, countably additive scalarmeasures on a Borel σ -algebra of subsets of a compact set K . We equip thisspace with the norm k m k rca( K ) := | m | ( K ). Here |·| stands for the total variationof the measure m . Then the pair (rca( K ) , k · k rca( K ) ) is a Banach space. TheRiesz representation theorem states that in this case rca( K ) is conjugate spaceto the space (cid:0) C ( K ) , k · k ∞ (cid:1) .Let ( X, A , ρ ) be a measure space, Y – a Banach space and denote by h· , ·i the duality pairing. Recall, that a function f : X → Y ∗ is weakly ∗ -measurable if for any y ∈ Y the function x → h f ( x ) , y i is A -measurable.The elements of a space L ∞ w ∗ ( Ω ; rca( K )) are the functions ν : Ω ∋ x → ν ( x ) ∈ rca( K ) , that are weakly ∗ -measurable and such thatess sup (cid:8) k ν ( x ) k rca( K ) : x ∈ Ω (cid:9) < + ∞ , whereess sup (cid:8) k ν ( x ) k rca( K ) : x ∈ Ω (cid:9) := inf (cid:8) α ∈ R ∪{∞} : k ν ( x ) k rca( K ) ≤ α a.e. in Ω (cid:9) . We endow this space with a norm k ν k L ∞ w ∗ ( Ω, rca( K )) := ess sup (cid:8) k ν ( x ) k rca( K ) : x ∈ Ω (cid:9) . It is proved in [19] that the space (cid:0) L ( Ω, C ( K )) (cid:1) ∗ is isometrically isomorphicto the space L ∞ w ∗ ( Ω ; rca( K )).The set Y ( Ω, K ) of the
Young measures consists of those functions fromthe space (cid:0) L ∞ w ∗ ( Ω ; rca( K )) , k ν k L ∞ w ∗ ( Ω, rca( K )) (cid:1) , whose values are probability mea-sures on K . The Theorem 3.1.6 allows one to conclude that for any measurablefunction f : Ω → K there exists a Young measure ν f associated with this func-tion.A particularly simple, yet important, example of a Young measure is a homo-geneous Young measure . This is a case, when ’the family of probability measures ν = ( ν x ) x ∈ Ω reduces to a unique single measure: ν x = ν for a.e x ∈ Ω ’, to cite onvergence of the sequences of Young measures Y ( Ω, K ) is a subset of the set of proba-bility measures on K . Majority of the specific examples of the Young measuresthat can be found in the literature are homogeneous ones, see for example [6],Example 3.44; [12], paragraphs 3.2, 3.3, 4.6, 6.2; [13], second and third para-graph of the first chapter, second and fourth paragraphs of chapter 9; [19] pages116, 117.Let Ω ⊂ R d be a bounded open set with the Lebesgue measure dx and let M > Ω . Define dλ ( x ) := M dx and consider acompact set K ⊂ R l with the Lebesgue measure dµ and a Borel measurablefunction f : Ω → K . Due to the fact that homogeneous Young measures are’one-element families’, we will write ’ ν ’ instead of ’ ν = ( ν x ) x ∈ Ω ’ for such mea-sures and denote by ’ ν f ’ the homogeneous Young measure associated with thefunction f .According to the Convention 3.1.1, the Theorem 3.1.6 in [19] and the The-orem 3.1 in [15] we will use the following definition of the homogeneous Youngmeasure. Definition 2.1. (i) we say that a mapping ν ∈ Y ( Ω, K ) is a homogeneousYoung measure if it is constant on Ω ;(ii) let ν f be a Young measure associated with a Borel function f : Ω → K .We say that ν f is a homogeneous Young measure if it is constant on Ω and is an image of the measure λ under f , i.e. ν f = λ ◦ f − . Let Ω ⊂ R d be a nonempty, bounded open set of the Lebesgue measure M > { Ω } – an open partition of Ω into at most countable number of opensubsets Ω , Ω , . . . , Ω n , . . . such that(i) the elements of { Ω } are pairwise disjoint;(ii) S i Ω i = Ω , where A denotes the closure of the set A .Let us consider functions f i : Ω i → K ⊂ R d , i = 1 , , ... , with inverses f − i that are continuously differentiable on f ( Ω i ) and let K i := f i ( Ω i ) be compact.Denote for each i = 1 , , ... the Jacobian of f − i by J f − i . Definition 3.1.
We say that a function f : Ω → K , with K := f ( Ω ) compact,is an m-oscillating function if it is of the form f ( x ) = X i f i ( x ) χ Ω i ( x ) . (3.1) Remark 3.1.
The letter ’m’ in the above definition refers to the fact, that inone dimensional case both the functions f i and f − i are strictly monotonic ones.This is to distinguish from the case when oscillating functions are piecewiseconstant, for example the Rademacher functions. onvergence of the sequences of Young measures Theorem 3.1.
The Young measure associated with function f satisfying (3.1)is a homogeneous one and its density g with respect to the Lebesgue measure on K is given by the formula g ( y ) = 1 M X i : y ∈ K i | J f − i ( y ) | . (3.2)The above result suggests introducing the following notion. Definition 3.2.
Let an oscillating function f be given by the equation (3.1).The total slope Jt f of f is defined by Jt f ( y ) := X i : y ∈ K i | J f − i ( y ) | . Example 3.1. (i) consider an example on page 117 in [19] of a sequence ( f n ) of m-oscillating nonperiodic functions, where for each n ∈ N we have f n ( x ) := (cid:0) x ( n + k − − k + 1 (cid:1) n + kn , x ∈ (cid:16) k − n + k − , kn + k (cid:17) , k ∈ N odd (cid:0) k − x ( n + k ) (cid:1) n + k − n , x ∈ h k − n + k − , kn + k (cid:17) , k ∈ N even . Then the sequence ( Jt f n ) of the respective total slopes is constant witheach element equal to 1. It means that the Young measure associated witheach f n (and therefore the Young measure generated by the sequence ( f n ) )is a homogeneous one, absolutely continuous with respect to the Lebesguemeasure on [0 , with density that is equal to 1 a.e;(ii) let f n ( t ) = sin(2 nπt ) , t ∈ (0 , , n ∈ N . Then ∀ n ∈ N Jt f n ( y ) = π √ − y , and thus the Young measure associated with each f n is homogeneous andabsolutely continuous with respect to the Lebesgue measure. Its density isequal to Jt f n . The following result follows from Theorem 1.64 in [6] and Theorem 3.1.
Theorem 3.2.
Consider the set A of m-oscillating functions defined on anonempty, bounded open set Ω ⊂ R d of positive Lebesgue measure, having valuesin a compact set K ⊂ R d . Then the set of the Young measures associated withthe elements of A is relatively weakly compact. We will now recall classical theorems concerning weak sequential convergence offunctions and measures that will be needed in the sequel. The expression ’weak onvergence of the sequences of Young measures K )’ (with thetotal variation norm).Recall that if ρ is a measure on K and for some function w : K → R integrablewith respect to the measure ξ there holds: for any Borel subset A of K we have ρ ( A ) = R A w ( y ) dξ ( y ), then the function w is called a density of the measure ρ .In this case ρ is absolutely continuous with respect to ξ (shortly: ξ -continuous): ξ ( A ) = 0 ⇒ ρ ( A ) = 0.Let ( X, A , ρ ) be a measure space and consider a sequence ( u n ) of scalarfunctions defined on X and integrable with respect to the measure ρ (that is, ∀ n ∈ N f n ∈ L ρ ( X )) and a function u ∈ L ρ ( X ). Recall that ( u n ) convergesweakly sequentially to u if ∀ g ∈ L ∞ ρ ( X ) lim n →∞ Z X u n gdρ = Z X ugdρ. The following theorem characterizes weak sequential L convergence of functionsand weak convergence of measures. We refer the reader to [6, 3, 5]. Theorem 4.1. (a) (J. Dieudonn´e, 1957) let X be a locally compact Hausdorffspace and ( X, A , ρ ) – a measure space with ρ regular. A sequence ( u n ) ⊂ L ρ ( X ) converges weakly to some u ∈ L ρ ( X ) if and only if ∀ A ∈ A thelimit lim n →∞ Z A u n dρ exists and is finite;(b) let X be a locally compact Hausdorff space and denote by B ( X ) the σ -algebra of Borel subsets of X . A sequence ( ρ n ) of scalar measures on B ( X ) converges weakly to some scalar measure ρ on B ( X ) if and only if ∀ A ∈ B ( X ) the limit lim n →∞ ρ n ( A ) exists and is finite. Corollary 4.1. (a) let ( ρ n ) be a sequence of measures having respective den-sities u n , n ∈ N . Then the sequence ( u n ) is weakly convergent in L ( X ) to some function h if and only if the sequence ( ρ n ) is weakly convergentto some measure η ;(b) assume additionally, that X ⊂ R l is compact and let ( ρ n ) be a sequenceof homogeneous Young measures having respective densities u n . Then thesequence ( u n ) is weakly convergent in L ( X ) to some function h if andonly if the sequence ( ρ n ) is weakly convergent to some measure η . We now introduce the notion of a density of a Young measure. We remember,that Young measure is in fact a family ν = ( ν x ) x ∈ Ω . In a special case, when onvergence of the sequences of Young measures x ) we say about homogeneous Young measure.Let Ω be a nonempty, bounded open subset of R n , K – compact subset of R l . The Borel σ -algebra of subsets of K will be denoted B ( K ). Definition 4.1.
We say that a family h = ( h x ) x ∈ Ω is a density of a Youngmeasure ν with respect to the measure ξ defined on B ( K ) if for any x ∈ Ω thefunction h x is a density of the measure ν x i.e. for any A ∈ B ( K ) there holds ν x ( A ) = R A h x ( y ) dξ ( y ) . The next result follows directly from the above definition.
Proposition 4.1.
Let ν be a Young measure and let h = ( h x ) x ∈ Ω be a densityof ν with respect to the measure ξ . Then ν is a homogeneous Young measure ifand only if the family h = ( h x ) x ∈ Ω consists of one element only, up to the setof ξ -measure . Example 4.1. (i) consider a function f : (0 , → [0 , given by the formula f ( x ) := ( x, if x ∈ (cid:0) , (cid:1) if x ∈ (cid:2) , (cid:1) . Then the Young measure associated with f has no density;(ii) the density of the Young measure associated with each element of the se-quence ( f n ) , where f n ( x ) := sin(2 nπx ) , x ∈ (0 , , is a ’one elementfamily’ h = ( h x ) x ∈ Ω , with h ( y ) = π √ − y ;(iii) let Ω = (0 , , K = [0 , and denote by µ a Lebesgue measure on B ( K ) .Consider a family h = ( h x ) x ∈ Ω of functions defined as follows: for each x ∈ Ω and y ∈ Kh x ( y ) := ( x y if y ∈ [0 , x ) − − x y + − x if y ∈ [ x, . Then h = (2 h x ) x ∈ Ω is a density of a nonhomogeneous Young measure ν = ( ν x ) x ∈ Ω , where for each x ∈ Ω we have ν x := 2 h x µ ( dy ) ;(iv) for this generalization of the above example we need some of the basicnotions of the set-valued analysis. All the necessary set-valued theory canbe found, for example, in [4, 7, 10, 18].Let Ω and K be given nonempty sets. By a multivalued mapping or mul-tifunction we mean a mapping T : Ω → K , often denoted by T : Ω K .For given set A ⊂ K denote T − ( A ) := { x ∈ Ω : T ( x ) ∩ A = ∅} . We saythat a multifunction T is lower semicontinuous at x ∈ Ω if and only iffor every open set V ⊂ K such that x ∈ T − ( V ) , x is an interior pointof T − ( V ) . A selection of a multifunction T is a single valued function onvergence of the sequences of Young measures t : Ω → K such that for all x ∈ Ω there holds t ( x ) ∈ T ( x ) . By the Axiomof Choice every multifunction has a selection, but one of the main prob-lems of set-valued analysis is investigation of the existence of selectionshaving certain regularity properties, like measurability or continuity.We say that a set D ⊂ L ( K ) is decomposable , if for all u, v ∈ D and all A ∈ B ( K ) the function χ A · u + χ K \ A · v is an element of D ; the symbol χ A stands for the characteristic function of the set A .Let Ω = [0 , , K = [0 , , µ – a Lebesgue measure on B ( K ) and let T : Ω L ( K ) be a lower semicontinuous multifunction having nonempty,closed and decomposable values. By the Bressan-Colombo-Fryszkowski the-orem, see for instance [7], Theorem 42 or [18], Theorem (7.18), there e-xists a continuous selection for T , that is a continuous function t : Ω → L ( K ) such that for each x ∈ Ω there holds t ( x ) ∈ T ( x ) . The family h = ( h x ) x ∈ Ω , where h x := t ( x ) and t ( x ) ∈ L ( K ) is µ -a.e. nonzero, givesrise to the density of the nonhomogeneous Young measure. We will now recall two classical theorems that will be needed in the sequel.We again refer the reader to [6, 3, 5].
Theorem 4.2. (a) ( Radon-Nikodym theorem for finite measures ) let µ and ρ be finite measures on a measurable space ( X, A ) and assumethat ρ is µ -continuous. Then there exists a unique ( up to the set of µ -measure µ -integrable function w ∈ L ( X ) such that ∀ A ∈ A ρ ( A ) = Z A wdµ ; (b) ( Vitali-Hahn-Saks theorem ) let ( X, A ) be measurable space, µ – a nonnegative finite measure and let ( ρ n ) be a sequence of µ -continuous scalar measures on A . If for any A ∈ A the limit lim n →∞ ρ n ( A ) exists, then the formula: ∀ A ∈ A ρ ( A ) := lim n →∞ ρ n ( A ) defines a µ -continuous scalar measure on A . Let the family ( h x ) x ∈ Ω be a density of a Young measure ( ν x ) x ∈ Ω . We thenhave the following theorem. onvergence of the sequences of Young measures Theorem 4.3.
Let ( Ω, dist) be a metric space and assume that the mapping h : Ω ∋ x → h ( x ) := h x ∈ L ( K ) is ( Ω, dist) – weakly-sequentially-in- L ( K ) continuous. Let ( x n ) be a sequencein Ω convergent to x ∈ Ω . Then the sequence ( h x n ) converges weakly to h x if and only if the sequence ( ν x n ) converges weakly to the measure ν x havingdensity h x .Proof. The first part of the theorem follows from part (b) of the Corollary 4.1.Now let the sequence ( ν x n ) converges weakly to the measure η . It follows fromthe Vitali-Hahn-Saks theorem that the measure η is µ -continuous, so by theRadon-Nikodym theorem it has a density r . Choose and fix set A ∈ B ( K ). Thefact that r = h x up to the set of µ -measure 0 follows from arbitrariness of thechoice of A and the inequality (cid:12)(cid:12)(cid:12) Z A r ( y ) dµ − Z A h x ( y ) dµ (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Z A r ( y ) dµ − η ( A ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) η ( A ) − ν x n ( A ) (cid:12)(cid:12)(cid:12) ++ (cid:12)(cid:12)(cid:12) ν x n ( A ) − Z A h x n ( y ) dµ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z A h x n ( y ) dµ − Z A h x n ( y ) dµ (cid:12)(cid:12)(cid:12) , which in turn implies that η = ν x .The next result follows from the above theorem and weak sequential com-pletness of L ( K ) (Theorem 1.3.13 in [11]). Corollary 4.2.
The density h = ( h x ) x ∈ Ω of a Young measure ν = ( ν x ) x ∈ Ω isa weakly sequentially closed set. We now turn to the case of homogeneous Young measures associated withBorel functions. Denote by µ the Lebesgue measure on B ( K ), by M – theset of all Borel measurable functions from Ω to K , by M H ⊂ M – the set offunctions such that the Young measures associated with them are homogeneousand µ -continuous, and define • F := (cid:8) ν h ∈ Y ( Ω, K ) : h ∈ M H (cid:9) ; • D := (cid:8) u : K → R : u is the density of the measure from F (cid:9) . Recall that for convex subsets of a normed space strong and weak closurescoincide, see for example Proposition A.3.19 in [8]. We thus have the followingtheorem.
Theorem 4.4.
Assume that the set K is convex. Then the set F is closed inthe norm topology of (cid:0) rca( K ) , k · k rca( K ) (cid:1) if and only if the set D is closed in thenorm topology of the space L ( K ) . onvergence of the sequences of Young measures Proof.
We use part (b) of the Corollary 4.1. The set K is convex so are M H , F and D . Assume that the set F is weakly closed and let u n be a sequence from D convergent weakly to u . Then the sequence ( ν n ), where ν n = ν u n = u n dµ ,converges weakly to ν ∈ F . Assumptions that the density w of ν is not equal µ -almost everywhere to u and that u / ∈ D lead to a contradiction, provingthat the set D is weakly closed in L ( K ), hence strongly closed.Conversely, assume that the set D is weakly closed and let ( ρ n ) be a sequenceof the Young measures from the set F , weakly convergent to the measure η . De-note their respective densities by u n . Again, from the Vitali-Hahn-Saks theoremand the Radon-Nikodym theorem we infer respectively, that the measure η is µ -continuous and has a density r . The fact that r = h up to the set of µ -measure0 is proved analogously like in the Theorem 4.3, while the inequality (cid:12)(cid:12)(cid:12) − Z K hdµ (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) − Z K u n dµ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z K u n dµ − Z K hdµ (cid:12)(cid:12)(cid:12) shows, that η is a probability measure on K .The uniqueness of the weak limit gives the weak ∗ -measurability of the map-ping Ω ∋ x → η ∈ rca( K ) , proving that η is a Young measure. Its homogenity is a consequence of theProposition 4.1.Respective weak convergences of Young measures and their densities stillhold without convexity assumptions on K . Corollary 4.3.
Let K ⊂ R l be compact. Let ( ρ n ) be a sequence of µ -continuoushomogeneous Young measures with respective densities u n .Then the sequence ( u n ) is weakly convergent in L ( K ) to some function h if andonly if the sequence ( ρ n ) is weakly convergent in the Banach space (rca( K ) , k · k rca( K ) ) to a homogeneous Young measure η with density h . Consider again examples from pages 116 and 117 in [19] of the sequencesof m-oscillating functions and the Young measures they generate. In [14] it isshown, that the Young measures associated with each element of the particularsequence form constant, hence trivially weakly ∗ convergent, sequence. It fol-lows from the fact, that the sequence of the total slopes of these m-oscillatingfunctions is constant. It is a special case of a more general situation describedin the next result. It originates from a simple observation in [17]. Theorem 4.5.
Consider the sequence ( f n ) of m-oscillating functions. Assumefurther that the sequence ( Jt f n ) of respective total slopes is monotonic. Thenthe sequence ( ρ n ) of Young measures associated with the functions f n is weaklyconvergent to the homogeneous Young measure ρ . Moreover, the measure ρ hasdensity which is equal to the weak L limit of the sequence ( u n ) of densities ofthe Young measures associated with the functions f n . onvergence of the sequences of Young measures Proof.
We can assume that the sequence ( Jt f n ) is nondecreasing. By Theorem 3.1the Young measures ρ n are homogeneous with densities given by the equation (3.2).Then for any m, n ∈ N , m ≤ n and any A ∈ B ( K ) there holds an inequality Z A Jt f m dµ ≤ Z A Jt f n dµ, which, together with Theorem 4.1 and Corollary 4.3, gives the result. References [1] H. Attouch, G. Buttazzo and G. Michaille,
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