CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE

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1 ESAIM: COCV 22 (206) DOI: 0.05/cocv/ ESAIM: Contol, Optmsaton and Calculus of Vaatons CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE Baboa Benešová,2 and Matn Kužík,2,3 Abstact. We chaacteze Young measues geneated by gadents of b-lpschtz oentatonpesevng maps n the plane. Ths queston s motvated by vaatonal poblems n nonlnea elastcty whee the oentaton pesevaton and njectvty of the admssble defomatons ae key equements. These esults enable us to deve new weak lowe semcontnuty esults fo ntegal functonals dependng on gadents. As an applcaton, we show the exstence of a mnmze fo an ntegal functonal wth nonpolyconvex enegy densty among b-lpschtz homeomophsms. Mathematcs Subject Classfcaton. 49J45, 35B05. Receved May 3, 204. Revsed Novembe 4, 204. Publshed onlne Januay 28, Intoducton The am of ths pape s to descbe oscllatoy popetes of sequences of gadents of b-lpschtz maps n the plane that peseve the oentaton,.e., the gadents of whch have a postve detemnant. Such mappngs natually appea n non-lnea hypeelastcty whee they act as defomatons. Although thee ae moe geneal defntons of a defomaton,.e. a functon y : R n that maps each pont n the efeence confguaton to ts cuent poston, we confne ouselves to the one by Calet ([9], p. 27) whch eques njectvty n the doman R n, suffcent smoothness and oentaton pesevaton. Hee, suffcent smoothness wll mean that a consdeed defomaton wll be a homeomophsm n ode to pevent cacks o cavtaton and ts (weak) defomaton gadent wll be ntegable,.e. y W,p (; R n )wth<p +. Clealy, a defomaton s an nvetble map but, n ou modelng, we put an addtonal equement on y namely, t should agan qualfy as a defomaton, whch s motvated by the fact that we am to model the elastc esponse of the specmen. In the elastc egme, the specmen etuns to ts ognal shape afte all loads ae eleased and so, snce the ôles of the efeence and the defomed confguaton can be exchanged, we would lke to undestand the eleasng of loads as applyng a new loadng, nvese to the ognal one, n the defomed confguaton and the etun of the specmen as the coespondng defomaton. Thus, we defne the followng Keywods and phases. Oentaton-pesevng mappngs, Young measues. Ths wok was suppoted by the GAČR Gants P20/0/0357, P20/2/067, P07/2/02, S, and S. Depatment of Mathematcs I, RWTH Aachen Unvesty, Aachen, Gemany. 2 Insttute fo Mathematcs, Unvesty of Wüzbug, Eml-Fsche-Staße 40, Wüzbug, Gemany. baboa.benesova@mathematk.un-wuezbug.de 3 Insttute of Infomaton Theoy and Automaton, Czech Academy of Scences, Pod vodáenskou věží4,cz-8208paha8, Czech Republc & Faculty of Cvl Engneeng, Czech Techncal Unvesty, Thákuova 7, Paha 6, Czech Republc. kuzk@uta.cas.cz Atcle publshed by EDP Scences c EDP Scences, SMAI 206

2 268 B. BENEŠOVÁ AND M. KRUŽÍK set of defomatons W,p, p + (; R n )= { y : y() an oentaton pesevng homeomophsm; y W,p (; R n )andy W,p (y(); R n ) }. (.) Although nvetblty of defomatons s a fundamental equement n elastcty t s stll often omtted n modelng due to the lack of appopate mathematcal tools to handle t. Howeve, let us menton that some deas of ncopoatng nvetblty of the defomaton aleady appeaed e.g. n [4, 0, 5, 8, 9, 27, 28, 32] and vey ecently e.g. n [4, 20]. Stable states of the specmen ae found by mnmzng J(y) = W ( y(x)) dx, (.2) whee W : R n n R s the stoed enegy densty,.e. the potental of the fst Pola Kchhoff stess tenso, ove the set of admssble defomatons (.); possbly wth espect to a Dchlet bounday condton y = y 0 on. A natual, stll open, queston s unde whch mnmal condtons on a contnuous W satsfyng W (A) =+ f det A 0and W (A) + wheneve det A 0 + (.3) we can guaantee that J s weakly lowe-semcontnuous on (.). In fact, Poblem n Ball s pape [6]: Pove the exstence of enegy mnmzes fo elastostatcs fo quasconvex stoed-enegy functons satsfyng (.3) s closely elated. Hee we answe ths queston fo the specal case of b-lpschtz mappngs n the plane;.e. weestctou attenton to the settng p =,n = 2. It s natual to conjectue that the sought equvalent chaactezaton of weak* lowe semcontnuty wll lead to a sutable noton of quasconvexty. We confm ths conjectue and show that J s weakly* lowe semcontnuous on W,, + (; R 2 ) f and only f t s b-quasconvex n the sense of Defnton 3.. Remak. (Quasconvexty). We say that W s quasconvex f W (A) W (A + ϕ(x)) dx (.4) holds fo all ϕ W, 0 (; R n )andalla R n n [25]. It s well-known [2] thatfw takes only fnte values and s quasconvex then J n (.2) s weakly* lowe semcontnuous on W, (; R n ) and so, n patcula, also on W,, + (; R 2 ). Nevetheless, as we shall see, classcal quasconvexty s too estctve n the b-lpschtz settng; ndeed, snce we naowed the set of defomatons t can be expected that a lage class of eneges wll lead to weak* lowe semcontnuty of J. Ths can be also undestood fom a mechancal pont of vew: quasconvex mateals ae descbed by eneges havng the popety that among all defomatons wth affne bounday data the affne ones ae stable. Thus, snce we now estcted the set of defomatons t seems natual to vefy (.4) only fo b-lpschtz functons; ths s ndeed the sought afte convexty noton whch we call b-quasconvexty (cf. Def. 3.). To pove ou man esult, we completely and explctly chaacteze gadent Young measues geneated by sequences n W,, + (; R 2 )(cf. Sect. 3). Young measues extend the noton of solutons fom Sobolev mappngs to paametzed measues [5,7,29 3,33,34,36]. The dea s to descbe the lmt behavo of {J(y k )} k N along a mnmzng sequence {y k } k N. Actually, one needs to wok wth the so-called gadent Young measues because t s the gadent of the defomaton enteng the enegy n (.2). The explct chaactezaton s due to Kndelehe and Pedegal [2, 22]; howeve, t does not take nto account any constant on detemnants o nvetblty of the geneatng mappngs. In spte of ths dawback, gadent Young measues ae massvely used n lteatue to model sold-to-sold phase tanstons as appeang n, e.g., shape memoy alloys; cf. [7, 24, 26, 29, 30].

3 CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE 269 Yet, not excludng matces wth a negatve detemnant may add non-ealstc phenomena to the model. Indeed, t s well-known that the modelng of sold-to-sold phase tanstons va Young measues s closely elated to the so-called quasconvex envelope of W whch must be convex along ank-one lnes,.e. lnes whose elements dffe by a ank-one matx. Not excludng matces wth negatve detemnants, howeve, adds many non-physcal ank-one lnes to the poblem. Notce, fo nstance, that any element of SO(2) s on a ank-one lne wth any element of O(2) \ SO(2). Consequently, the detemnant must nevtably change ts sgn on such lne. The fst attempt to nclude constants on the sgn of the detemnant of the geneatng sequence appeaed n [2] whee quas-egula geneatng sequences n the plane wee consdeed; howeve njectvty of the mappngs could only be teated n the homogeneous case. Then, n [8] the chaactezaton of gadent Young measues geneated by sequences whose gadents ae nvetble matces fo the case whee gadents as well as the nvese matces ae bounded n the L -nom was gven. Vey ecently, Koumatos et al. [23] chaactezed Young measues geneated by oentaton pesevng maps n W,p fo <p<n; howeve they dd not account fo the estcton that defomatons should be njectve. Theefoe, ths contbuton (to ou best knowledge) pesents the fst chaactezaton of Young measues that ae geneated by sequences that ae oentaton-pesevng and globally nvetble andsoqualfytobe admssble defomatons n elastcty. Geneally speakng, the man dffculty n chaactezng sets of Young measues geneated by defomatons (o, at least, mappngs havng constants on the nvetblty and/o detemnant of the defomaton gadent) s that ths constant s non-convex. Thus, many of the standadly used technques such as smoothng by a mollfe kenel ae not applcable. In ou context, we need to be able to modfy the geneatng sequence on a vanshngly small set nea the bounday to have the same bounday condtons as the lmt;.e. to constuct a cut-off technque. It can be seen fom (.4), that standad poofs of chaactezatons of gadent Young measues [2,22] o weak lowe semcontnuty of quasconvex functonals [2] wll ely on such technques snce the test functons n (.4) have fxed bounday data. Usually, the cut-off s ealzed by convex aveagng whch s, of couse, uled out hee. Novel deas n [8, 23] ae to solve dffeental nclusons nea the bounday to ovecome ths dawback. Ths allows to mpose estctons on the detemnant of the geneatng sequence n seveal soft-egmes ; nevetheless, such technques have not been genealzed to moe gd constants lke the global nvetblty. Hee we follow a dffeent appoach and, fo b-lpschtz mappngs n the plane, we obtan the esult by explotng b-lpschtz extenson theoems [3, 35]. Thus, by followng a stategy nsped by [4] wemodfythe geneatng sequence (on a set of gadually vanshng measue nea the bounday) fst on a one-dmensonal gd and then extend t. The man eason why we confne ouselves to the b-lpschtz case and do not wok n W,p, p + (; R 2 )wthp< s the fact that ou technque eles on the extenson theoem o, n othe wods, a full chaactezaton of taces of b-lpschtz functons. To ou best knowledge, such a chaactezaton s at the moment completely open n W,p, p + (; R 2 )wthp<. Stll, let us pont out ts mpotance fo fndng mnmzes of J ove (.): n fact, constuctng an extenson theoem allows to pecsely chaacteze the set of Dchlet bounday data admssble fo ths poblem. Notce that ths queston appeas also n the exstence poof fo polyconvex mateals and usually one assumes thee that the set of admssble defomatons s nonempty; [9]. Remak.2 (Gowth condtons). Even though n ths pape we estct ou attenton to b-lpschtz functons, let us pont unde whch gowth of the enegy we can guaantee that the mnmzng sequence of J les n W,p, p + (; R n ). Namely, t follows fom the woks of Ball [3, 4] that t suffces to eque that W s fnte only on the set of matces wth postve detemnant and ( cof stands fo the cofacto n dmenson 2 o 3) ( C A p + ) ( det A + cof(a) p det A p W (A) C A p + ) det A + cof(a) p +, (.5) det Ap as well as fx sutable bounday data (fo example b-lpschtz ones) 4. 4 As ponted above, snce the taces of functons n W,p, p + (; R 2 ) ae not pecsely chaactezed to date, t s had to decde what sutable bounday data ae. In any case, n the plane b-lpschtz bounday data ae suffcent.

4 270 B. BENEŠOVÁ AND M. KRUŽÍK Polyconvexty,.e. convexty n all mnos of A, s fully compatble wth such gowth condtons (they ae themselves polyconvex) whence f W s polyconvex mnmzes of (.2) ove W,p (; R n ), p>nae ndeed defomatons;.e. ae globally nvetble and elements of W,p, p + (; R n ). We efe, e.g., to[9, 2] fo vaous genealzatons of ths esult. Howeve, whle polyconvexty s a suffcent condton t s not a necessay one. On the othe hand, classcal esults on quasconvexty yeldng exstence of mnmzes [2] ae compatble wth nethe the gowth condtons poposed n ths emak no (.3). In fact, exstence of a mnmze of (.2) on W,p (; R n ) fo quasconvex W can be, to date, poved only f c ( + A p ) W (A) c ( + A p ). (.6) The eason why the cuent poofs of exstence of mnmzes fo quasconvex cannot be extended to (.5) s exactly the non-convexty detaled above. The plan of the pape s as follows. We fst ntoduce necessay defntons and tools n Secton 2. Then we state the man esults n Secton 3. Poofs ae postponed to Secton 4 whle the novel cut-off technque s pesented n Secton Pelmnaes Befoe statng ou man theoems n Secton 3, let us summaze, at ths pont, the notaton as well as backgound nfomaton that we shall use late on. We defne the followng subsets of the set of nvetble matces: Rϱ 2 2 = { A nvetble; A ϱ & A ϱ }, (2.) Rϱ+ 2 2 = { A Rϱ 2 2 ;deta>0 } (2.2) fo ϱ<. Note that both Rϱ 2 2 and Rϱ+ 2 2 ae compact. Set nv = ϱ ϱ = ϱ ϱ+. We assume that the matx nom used above s sub-multplcatve,.e. that AB A B fo all A, B and such that the nom of the dentty matx s one. Ths means that f A ϱ+ then mn( A, A ) /ϱ. Defnton 2.. A mappng y : R 2 s called L-b-Lpschtz (o shotly b-lpschtz) f thee s L such that fo all x,x 2 L x x 2 y(x ) y(x 2 ) L x x 2. (2.3) The numbe L s called the b-lpschtz constant of y. L Ths means that y as well as ts nvese y ae Lpschtz contnuous, hence y s homeomophc. Notce that y(x) L fo almost all x. Defnton 2.2. We say that {y k } k N W,, + (; R 2 ) s bounded n W,, + (; R 2 ) f the b-lpschtz constants of y k, k N, ae unfomly bounded and {y k } k N s bounded n W, (; R n ). Moeove, we say that y k yn W,, + (; R 2 ) f the sequence s bounded and y k yn W, (; R 2 ). We would lke to stess the fact that W,, + (; R 2 ) s not a lnea functon space.

5 CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE 27 Remak 2.3. Notce that f y k yn W,, + (; R 2 ), we can gve a pecse statement on how the nveses of {y k } convege f the taget doman s fxed thoughout the sequence;.e. f y k : fo all k N. Ths can be acheved fo example by fxng Dchlet bounday data though the sequence. In such a case t s easy to see that y k y n W, (,): Snce the gadents of the nveses y k ae unfomly bounded by the unfom b-lpschtz constants, we may select at least a subsequence convegng weakly* n W, (,) and thus stongly n L (,). Nevetheless, the latte allows us to pass to the lmt n the dentty y k (y k(x)) = x fo any x and theefoe to dentfy the weak* lmt as y ;nothewods, the weak lmt s dentfed ndependently of the selected subsequence whch assues that the whole sequence {y k } k N conveges weakly* to y. Let us now summaze the theoems on nvetblty, extenson fom the bounday n the b-lpschtz case and on appoxmaton by smooth functons needed below. Theoem 2.4 (Taken fom [4]). Let R n be a bounded Lpschtz doman. Let u 0 : R n be contnuous n and one-to-one n such that u 0 () s also bounded and Lpschtz. Let u W,p (; R n ) fo some p>n, u(x) =u 0 (x) fo all x, andletdet u >0 a.e. n. Fnally, assume that fo some q>n ( u(x)) q det u(x)dx<+. (2.4) Then u() = u 0 () and u s a homeomophsm of onto u 0 (). Moeove, the nvese map u W,q (u 0 (); R n ) and u (z) =( u(x)) fo z = u(x) and a.a. x. Remak 2.5. Let us pont out that the ognal statement of Theoem 2.4 eques that u 0 () satsfes the so-called cone condton and that s stongly Lpschtz. These condtons hold f and u 0 () ae bounded and Lpschtz domans (cf. [], pp ). Theoem 2.6 (Squae b-lpschtz extenson theoem due to [3] and pevously [35]). Thee exsts a geometc constant C such that evey L b-lpschtz map u : D(0, ) R 2 (wth D(0, ) the unt squae) admts a CL 4 b-lpschtz extenson v : D(0, ) Γ whee Γ s the bounded closed set such that Γ = u( D(0, )). Remak 2.7 (Rescaled squaes). Let us note, that the theoem above holds wth the same geometc constant C also fo escaled squaes D(0,ɛ)wthsomeɛ>0, possbly small. Indeed, fo u : D(0,ɛ) R 2, we defne the escaled functon ũ : D(0, ) R 2 though ũ(x) =ɛu(x/ɛ); note that both functons have the same b-lpschtz constant. Ths functon s then extended to obtan ṽ : D(0, ) R 2 as n the above theoem. Agan we escale ṽ, unde pesevaton of the b-lpschtz constant, to v : D(0,ɛ) R 2 v = ɛ ṽ(ɛx). So, v s CL4 b-lpschtz and, snce ũ concdes wth ṽ on the bounday of the unt squae, v concdes wth u on D(0,ɛ). Theoem 2.8 (Smooth appoxmaton [20] and n the b-lpschtz case also by [4]). Let R 2 be bounded open and y W,p (; R 2 )(<p< ) be an oentaton pesevng homeomophsm. Then t can be, n the W,p -nom, appoxmated by dffeomophsms havng the same bounday value as y. Moeove, f y s b- Lpschtz, then thee exsts a sequence of dffeomophsms {y k } havng the same bounday value as y and y k, y k appoxmate y, y n W,p -nom wth <p<, espectvely. 2.. Young measues We denote by ca(s) the set of Radon measues on a set S. Young measues on a bounded doman R n ae weakly* measuable mappngs x ν x : ca(r n n ) wth values n pobablty measues; the adjectve weakly* measuable means that, fo any v C 0 (R n n ), the mappng R : x ν x,v = R n n v(s)ν x (ds) s measuable n the usual sense. Let us emnd that, by the Resz theoem, ca(r n n ), nomed by the total vaaton, s a Banach space whch s sometcally somophc wth C 0 (R n n ),wheec 0 (R n n ) stands fo

6 272 B. BENEŠOVÁ AND M. KRUŽÍK the space of all contnuous functons R n n R vanshng at nfnty. Let us denote the set of all Young measues by Y(; R n n ). It s known (see e.g. [30]) that Y(; R n n ) s a convex subset of L w (;ca(r n n )) = L (; C 0 (R n n )), whee the subscpt w ndcates the afoementoned popety of weak* measuablty Let S R n n be a compact set. A classcal esult [33] states that fo evey sequence {Y k } k N bounded n L (; R n n ) such that Y k (x) S thee exsts a subsequence (denoted by the same ndces fo notatonal smplcty) and a Young measue ν = {ν x } x Y(; R n n ) satsfyng v C(S) : lm v(y k)= v(s)ν x (ds) weakly* n L (). (2.5) k R n n Moeove, ν x s suppoted on S fo almost all x. On the othe hand, f μ = {μ x } x, μ x s suppoted on S fo almost all x and x μ x s weakly* measuable then thee exst a sequence {Z k } k N L (; R n n ), Z k (x) S and (2.5) holdswthμ and Z k nstead of ν and Y k, espectvely. Let us denote by Y (; R n n ) the set of all Young measues whch ae ceated n ths way,.e. by takng all bounded sequences n L (; R n n ). Moeove, we denote by GY (; R n n ) the subset of Y (; R n n ) consstng of measues geneated by gadents of {y k } k N W, (; R n ),.e. Y k = y k n (2.5). The followng esult s due to Kndelehe and Pedegal [2, 22] (seealso[26, 29]): Theoem 2.9 (adapted fom [2, 22]). Let be a bounded Lpschtz doman. Then the paametzed measue ν Y (; R n n ) s n GY (; R n n ) f and only f () thee exsts z W, (; R n ) such that z(x) = R n n Aν x (da) fo a.e. x, (2) ψ( z(x)) R n n ψ(a)ν x (da) fo a.e. x and fo all ψ quasconvex, contnuous and bounded fom below, (3) supp ν x K fo some compact set K R n n fo a.e. x. 3. Man esults We shall denote, fo ϱ, GY, ( ϱ ; R 2 2 ) = { ν Y ( ; ) that ae geneated by ϱ-b-lpschtz, oentaton pesevng maps }, and GY, ( + ; R 2 2 ) = GY, ( ϱ ; R 2 2 ). ϱ As aleady ponted out n the ntoducton we seek fo an explct chaactezaton of GY, ( ) + ; R 2 2 ;tcan be expected that, when compaed to [2], we shall estct the suppot of the Young measue as n [2,8,23] but also alte the Jensen s nequalty by changng the noton of quasconvexty. Defnton 3.. Suppose v : R {+ } s bounded fom below and Boel measuable. Then we denote Zv(A) = nf v( ϕ(x)) dx, ϕ W,, A (;R 2 ) wth W,, ( A ; R 2 ) = {{ y W,, + ( ; R 2 ) ; y(x) =Ax f x } f det A>0, else. and say that v s b-quasconvex on f Zv(A) =v(a) fo all A R2 2. Hee we set nf =+.

7 CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE 273 Remak 3.2. () Notce that actually Zv(A) v(a) fdeta>0andzv(a) =+ othewse, so that Zv v n geneal. Moeove, the nfmum n the defnton of Zv(A) s, genecally, not attaned. (2) Any v as n Defnton 3. b-quasconvex f and only f v(a) v( ϕ(x)) dx (3.) fo all ϕ W,, + (; R 2 ), ϕ = Ax on and all A. Indeed, clealy f v s b-quasconvex then (3.) holds. On the othe hand, f (3.) holds, we have that v(a) Zv(A) foa by takng the nfmum n (3.). Moeove, Zv(A) v(a) fosucha, sothatzv(a) = v(a). (3) We ecall that the condton of b-quasconvexty s less estctve than the usual quasconvexty and thee obvously exst b-quasconvex functons on whch ae not quasconvex (fo example, take v : R R wth v(0) = and v(a) =0fA 0.). Also, we can allow fo the gowth (.3). (4) It s nteestng to nvestgate whethe, fo any v as fom Defnton 3., Zv(A) s aleady a b-quasconvex functon. If one wants to follow the standad appoach known fom the analyss of classcal quasconvex functon [2], ths conssts n showng that Zv can be actually eplaced by Z v defned though Z v(a) = nf v( ϕ(x)) dx, ϕ W,, A (;R 2 ) pecewse affne and that the latte s b-quasconvex. To do so, one eles on the densty of pecewse affne functon whch, n ou case, s avalable though Theoem 2.8. Moeove, to employ the densty agument, one needs to show that Z v s ank- convex on and hence contnuous. Ths s done by constuctng a sequence of faste and faste oscllatng lamnates that ae alteed nea the bounday to meet the bounday condton. Now, snce an appopate cut-off technque becomes avalable though ths wok, t seems that ths appoach should be feasble. Nevetheless, the detals ae beyond the scope of the pesent pape and we leave them fo futue wok. Let us emak that an altenatve to the above methods may be possble along the lnes of the ecent wok []. The man esult of ou pape s the followng chaactezaton theoem. Theoem ( 3.3. Let ) R 2 be a bounded Lpschtz doman. Let ν Y (; ). Then ν ; R 2 2 f and only f the followng thee condtons hold: GY, + ϱ s.t. supp ν x Rϱ+ 2 2 fo a.a. x, (3.2) u W,, + (; R 2 ) : u(x) = Aν x (da), (3.3) c(ϱ) >ϱsuch that fo a.a. x, all ϱ [ c(ϱ); + ], andallv O( ϱ) the followng nequalty s vald Zv( u(x)) v(a)ν x (da), (3.4) wth O(ϱ) = { v : R {+ }; v C ( Rϱ 2 2 ), v(a) =+ f A \ Rϱ+ 2 2 }. (3.5) An easy coollay s the followng: Coollay 3.4. Let R 2 be a bounded Lpschtz doman. Let v be n O(+ ). Let {y k } k N W,, + (; R 2 ) and suppose that y k y n W,, + (; R 2 ). Then v s b-quasconvex f and only f y I(y) = v( y(x)) dx s sequentally weakly* lowe semcontnuous wth espect to the convegence above.

8 274 B. BENEŠOVÁ AND M. KRUŽÍK Fnally, as an applcaton we can state the followng statement about the exstence of mnmzes. Poposton 3.5. Let R 2 be a bounded Lpschtz doman and let 0 v O(+ ) be b-quasconvex. Let futhe ε>0 and defne I ε : W,, + (; R 2 ) R I ε (u) = v( u(x)) dx + ε ( u L (; ) + u L (u();r )) 2 2. Let u 0 W,, ( ) + ; R 2 and { A = Then thee s a mnmze of I ε on A. Remak 3.6. u W,, + ( ; R 2 ) ; u = u 0 on }. () Note that, we needed n Theoem 3.3 that ϱ >ϱso that boundedness of v( y k)dx does not yeld the ght L -constant of the gadent of the mnmzng sequence. Ths s actually a known fact n the L - case [2] and s usually ovecome by assumng that the geneatng sequence does not need to be Lpschtz but s only bounded n some W,p (; R 2 ) space. Altenatvely, one can use Poposton 3.5 stated above. (2) It wll follow fom the poof that the constant c(ϱ) s actually detemned by the extenson Theoem 2.6. (3) Note that f one can show that Zv s aleady a b-quasconvex functon (cf. Rem.3.2(4)) then (3.4) canbe eplaced by equng that v( u(x)) v(a)ν x (da) (3.6) s fulflled fo all b-quasconvex v n O( ϱ). Indeed, (3.6) follows dectly fom (3.4) fv s b-quasconvex. On the othe hand, f (3.6) holds and f we knew that Zv s b-quasconvex, we know that Zv( u(x)) Zv(A)ν x (da) v(a)ν x (da), whee the second nequalty s due to Remak 3.2(). 4. Poofs Hee we pove Theoem 3.3. Actually, we follow n lage pats [2,29] snce, as ponted out n the ntoducton, the man dffculty les n constuctng an appopate cut-off whch we do n Secton 5; so, we mostly just sketch the poof and efe to these efeences. 4.. Poof of Theoem 3.3 necessty Condton (3.2) follows fom [8], Popostons 2.4 and 3.3 and fom the fact that any Young measue geneated by a sequence bounded n the L nom s suppoted on a compact set. In ode to show (3.3), ealze that t expesses the fact that the fst moment of ν s just the weak* lmt of a geneatng sequence { y k } L (; ). The sequence {y k } s also bounded n W,, + (; R 2 )and{y k } conveges stongly to some y W, (; R 2 ). Passng to the lmt n (2.3) wtten fo y k nstead of y shows that y s b-lpschtz. The L - weak* convegence of det y k to det y fnally mples that y W,, + (; R 2 )as a b-lpschtz map cannot change sgn of ts Jacoban on. To pove (3.4) we follow a standad stategy, e.g., as n [29]. Fst, we show that almost evey ndvdual measue ν x s a homogeneous Young measue geneated by b-lpschtz maps wth affne bounday data. The latte fact s mpled by Theoem 5.. Then(3.4) stems fom the vey defnton of b-quasconvexty.

9 CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE 275 Lemma 4.. Let ν GY, ϱ ( ) ; R 2 2.Thenμ = {ν a } x GY, ( ) ϱ ; R 2 2 fo a.e. a. Poof. Note that the constucton n the poof of ([29], Thm. 7.2) does not affect oentaton-pesevaton no the b-lpschtz popety. Namely, f gadents of a bounded sequence {u k } W,, + (; R 2 ) geneate ν then fo almost all a one constucts a localzed sequence {ju k (a + x/j)} j,k N (note that ths functon s clealy njectve f u k was; snce the nom of the gadent s just shfted ths yelds the b-lpschtz popety) whose gadents geneate μ as j, k. Poposton 4.2. Let ν GY, ( ) + ; R 2 2,suppν Rϱ 2 2 be such that y(x) = Aν ϱ x (da) fo almost all x, wheey W,, + (; R 2 ). Then fo all ϱ [ c(ϱ); + ], almostallx and all v O( ϱ) we have nv v(a)ν x (da) Zv( y(x)). (4.) Poof. We know fom Lemma 4. that μ = {ν a } x GY, ( ) ϱ ; R 2 2 fo a.e. a, sotheeextsts geneatng sequence { u k } k N such that {u k } k N W,, + (; R 2 ) and fo almost all x and all k N u k (x) Rϱ 2 2.Moeove,{u k } k N weakly* conveges to the map x ( y(a))x whch s b-lpschtz. Usng Coollay 5.2, we can, wthout loss of genealty, suppose that u k s ϱ-b-lpschtz fo all k N and u k (x) = y(a)x f x. Theefoe, we have v(a)ν a (da) = lm k 4.2. Poof of Theoem 3.3 suffcency v( u k (x)) dx Zv( y(a)). We need to show that condtons (3.2) (3.4) ae also suffcent fo ν Y (; ) to be n GY +, (; ). Put U ϱ A,, = {y WA (; R 2 ); y(x) Rϱ+ 2 2 fo a.a. x }; (4.2) In othe wods ths s the set of ϱ-b-lpschtz functons wth affne bounday values equal to x Ax. Consde fo A nv the set M ϱ A = {δ y; y U ϱ A }, (4.3) whee δ y ca( ) s defned fo all v C 0 ( )as δ y,v = v( y(x)) dx; Mϱ A wll denote ts weak closue. Lemma 4.3. Let A Rϱ Then the set Mϱ A s nonempty and convex. Poof. To show that M ϱ A s tval because x y(x) =Ax s an element of ths set as A has a postve detemnant. To show that M ϱ A s convex we follow ([29], Lem. 8.5). We take y,y 2 U ϱ A and, fo a gven λ (0, ), we fnd a subset D such that D = λ. Thee ae two countable dsjont famles of subsets of D and \ D of the fom {a + ɛ ; a D, ɛ > 0, a + ɛ D} and {b + ρ ; b \ D, ρ > 0, b + ρ \ D} such that D = (a + ɛ ) N 0, \ D = (b + ρ ) N,

10 276 B. BENEŠOVÁ AND M. KRUŽÍK whee the Lebesgue s measue of N 0 and N s zeo. We defne ( ) ( ɛ y x a ɛ + Aa f x a + ɛ, y x a ( ) ( y(x) = ρ y x b 2 ρ + Ab f x b + ρ, yeldng y(x) = y x b 2 Ax othewse, A ɛ ) ρ ) f x a + ɛ, f x b + ρ, othewse. We must show that y s ϱ-b-lpschtz; actually, as y(x) Rϱ+ 2 2 a.e., we only need to check the njectvty of the mappng. To ths end, we apply Theoem 2.4. Notcethat(2.4) clealy holds fo any q (, ) due to the a.e. bounds on y. Moeove, we have affne bounday data, y(x) =Ax, so that ndeed the bounday data fom a homeomophsm and, snce was a bounded Lpschtz doman, so wll be A = {Ax; x }. Thuswe conclude that, ndeed, y s ϱ-b-lpschtz. In patcula, y U ϱ A and δ y = λδ y +( λ)δ y2. The followng homogenzaton lemma can be poved the same way as ([29], Thm. 7.). The agument showng that a geneatng sequence of ν comes fom b-lpschtz oentaton pesevng maps comes fom Theoem 2.4 the same way as n the poof of Lemma 4.3. Lemma 4.4. Let {u k } k N W,, A (; R 2 ) be a bounded sequence n W,, + (; R 2 ). Let the Young measue ν GY, ( ) + ; R 2 2 be geneated by { u k } k N. Then thee s a anothe bounded sequence {w k } k N W,, A (; R 2 ) that geneates a homogeneous (.e. ndependent of x) measue ν defned though v(s) ν(ds) = v(s)ν x (ds)dx, (4.4) ϱ+ ϱ+ fo any v C(Rϱ+ 2 2 ) and almost all x. Moeove, ν GY, ( ) + ; R 2 2. Poposton 4.5. Let μ be a pobablty measue suppoted on a compact set K α+ fo some α and let A = K sμ(ds). Letϱ>αand let Zv(A) v(s)μ(ds), (4.5) fo all v O(ϱ). Thenμ GY, ( ) + ; R 2 2 and t s geneated by gadents of mappngs fom U ϱ A. K Poof. Fst, notce that A α < ϱ < +. Secondly, the set of measues μ n the statement of the poposton s convex and contans M ϱ A as ts convex and non vod subset due to Lemma 4.3. We show that no fxed μ satsfyng (4.5) can be sepaated fom the weak* closue of M ϱ A by a hypeplane. We ague by a contadcton agument. Then by the Hahn Banach theoem, assume that thee s ṽ C 0 ( ) that sepaates M ϱ A fom μ. In othe wods, thee exsts a constant c such that ν, ṽ c fo all ν M ϱ A and μ, ṽ < c. Howeve, snce we ae wokng wth pobablty measues, we may use ṽ c nstead of ṽ. Inthsway,wecan put c = 0. Hence, wthout loss of genealty, we assume that 0 ν, ṽ = ṽ(s)ν(ds) = ṽ( y(x)) dx, ϱ

11 CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE 277 fo all ν M ϱ A (and hence all y Uϱ A )and0> μ, ṽ. Now, the functon { ṽ(f ) f F Rϱ+ 2 2 v(f )=, + else, s n O(ϱ). Notce that t follows fom (4.5) thatzv(a) s fnte. Thus, Zv(A) =nf U ϱ v( y(x)) dx. A Hence, Zv(A) 0 and, by (4.5), 0 Zv(A) K v(s)μ(ds) = K ṽ(s)μ(ds). As ths holds fo all hypeplanes, μ M ϱ A, a contadcton. As C 0( ) s sepaable, the weak* topology on bounded sets n ts dual, ca( ), s metzable. Hence, thee s a sequence {u k } k N U ϱ A such that fo all v C(R2 2 ϱ+ )(andallv O(ϱ)) v( u k (x)) dx = v(s)μ(ds), (4.6) lm k and {u k } k N s bounded n W,, + (; ). Let ν be a Young measue geneated by { u k } (o a subsequence of t). Then we have fo v as above lm k v( u k (x)) dx = ϱ+ ϱ+ v(s)ν x (ds)dx = v(s)μ(ds). (4.7) ϱ+ As u k (x) =Ax fo x we apply Lemma 4.4 to get a new sequence {ũ k } bounded n W,, + (; ) wth ũ k (x) =Ax fo x. The sequence { ũ k } geneates a homogeneous Young measue ν gven by (4.4), so that n vew of (4.7) wegetfog L () lm k g(x)v( ũ k (x)) dx = g(x)dx ϱ+ v(s)ν x (ds)dx = ϱ+ g(x)v(s)μ(ds)dx. Lemma 4.6 (see [29], Lem. 7.9 fo a moe geneal case). Let R n be an open doman wth =0and let N be of the zeo Lebesgue measue. Fo k : \ N (0, + ) and {f k } k N L () thee exsts a set of ponts {a k } \ N and postve numbes {ɛ k }, ɛ k k (a k ) such that {a k + ɛ k } ae pawse dsjont fo each k N, = {a k + ɛ k } Nk wth N k =0and fo any j N and any g L () lm k f j (a k ) a k +ɛ k g(x)dx = f j (x)g(x)dx. In fact, the ponts {a k } can be chosen fom the ntesecton of sets of Lebesgue ponts of all f j, j N. Notce that ths ntesecton has the full Lebesgue s measue. Hee fo each j N, f j s dentfed wth ts pecse epesentatve ([6], p. 46). We adopt ths dentfcaton below wheneve we speak about a value of an ntegable functon at a patcula pont. PoofofTheoem3.3 suffcency. Some pats of the poof follow ([2], Poof of Thm. 6.). We ae lookng fo a sequence {u k } k N W,, + (; R 2 ) satsfyng lm v( u k (x))g(x)dx = v(s)ν x (ds)g(x)dx k n fo all g Γ and any v S, wheeγ and S ae countable dense subsets of C( ) andc(rϱ+ 2 2 ), espectvely. Fst of all notce that, as u W,, + (; R 2 )fom(3.3) s dffeentable n outsde a set of measue zeo called N, we may fnd fo evey a \ N and evey k>0some/k > k (a) > 0 such that fo any 0 <ɛ< k (a) we have fo evey y ɛ u(a + ɛy) u(a) ɛ u(a)y k (4.8)

12 278 B. BENEŠOVÁ AND M. KRUŽÍK Applyng Lemma 4.6 and usng ts notaton, we can fnd a k \ N, ɛ k k (a k ) such that fo all v S and all g Γ lm V (a k )g(a k ) ɛ k = V (x)g(x)dx, (4.9) k whee V (x) = v(s)ν x (ds). nv In vew of Lemma 4.5, weseethat{ν ak } x GY +, (; R n n ) s a homogeneous gadent Young measue and we call { yj k} j N W,, + (; R 2 ) ts geneatng sequence. We know that we can consde {yj k} j N U ϱ u(a k ) fo abtay + > ϱ >ϱ. Hence lm v( yj k (x))g(x)dx = V (a k ) g(x)dx (4.0) j and, n addton, yj k weakly conveges to the map x u(a k )x fo j n W, (; R 2 ) and due to the Azela-Ascol theoem also unfomly on C( ; R 2 ). Futhe, consde fo k N y k W, (a k + ɛ k ; R 2 ) defned fo x a k + ɛ k by ( ) x y k (x) :=u(a k )+ɛ k yj k ak whee j = j(k, ) wll be chosen late. Note that the above fomula defnes y k almost eveywhee n. Wewte fo almost evey x a k + ɛ k that ( ) u(x) y k (x) x u(x) u(a ak k) ɛ k u(a k ) ɛ k ( ) ( ) x ak x + ɛ k u(a k ) yj k ak 2ɛ k k, (4.) f j s lage enough. The fst tem on the ght-hand sde s bounded by ɛ k /k because of (4.8) whle the second one due to the unfom convegence of yj k x u(a k )x. Notcethaty k as well as u ae b- Lpschtz and oentaton pesevng on a k + ɛ k.ifx a k + ɛ k we set x =(x a k )/ɛ k and defne ũ( x) =ɛ k u(a k + ɛ k x) andỹ k ( x) =ɛ k y k(a k + ɛ k x) sothatwegetby(4.) fo all x ɛ k ũ( x) ỹ k ( x) 2 k Addtonally, note that the b-lpschtz constant of ỹ k, k N s agan L. Hence, we can take k > 0 lage enough that ũ ỹ k C( ;R 2 ) s abtaly small. Theefoe, we can use Theoem 5.2 and modfy ỹ k so that t has the same tace as ũ on the bounday of. Let us call ths modfcaton ũ k,.e., {ỹk ( x) f x, ũ k ( x) = ũ( x) othewse. Then we poceed n the opposte way to defne fo x = a k + ɛ k x: u k (x) =ɛ k ũ k ( x). Then, snce {u k } k N s bounded n W, (; R 2 ), we may assume the weak convegence of u k to u. It emans to show that evey u k s b-lpschtz. To do so, we agan apply Theoem 2.4. Weseethatfoevey k N det u k > 0. Futhe, sup k N ( u k ) < + follows fom constucton of the sequence, and u k = u on, sothatu k s ndeed b-lpschtz. ɛ k ɛ k

13 CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE 279 Fo k, fxed we take j = j(k, ) so lage that fo all (g, v) Γ S ɛ2 k g(a k + ɛ k y)v( u k j (y)) dy V (a k ) g(x)dx a k +ɛ k 2 k Usng ths estmate and (4.0) wegetfoany(g, v) Γ S lm g(x)v( u k (x)) dx = lm ɛ n k g(a k + ɛ k y)v( u k j (y)) dy k k = lm V (a k ) g(x)dx = V (x)g(x)dx k a k +ɛ k = v(s)ν x (ds)g(x)dx Poofs of Coollay 3.4 and Poposton 3.5 PoofofCoollay3.4. Fo showng the weak lowe semcontnuty, we ealze that the sequence { y k } k N geneates a measue n GY, ( ) + ; R 2 2 and so f v s b-quasconvex we easly have fom (3.4) v( y(x))dx = Zv( y(x))dx v(s)ν x (ds)dx = lm nf v( y k )dx. k On the othe hand, we ealze that evey y W,, A (; R 2 ) defnes a homogeneous Young measue ν GY, + (; ) by settng f(s)ν(ds) = f( y(x)) dx fo evey f contnuous on matces wth postve detemnant. Notce that the fst moment of ν s A. Let{ y k } k N be a geneatng sequence fo ν whch can be taken such that {y k } k N W,, A (; R 2 ). Moeove, the weak* lmt of y k s A. As we assume that I(y) = v( y(x)) dx and that I s weakly lowe semcontnuous on W,, A (; R 2 )weget whch shows that v s b-quasconvex. v(a) lm nf I(y k)= v(s)ν(ds)dx = k v( y(x)) dx, PoofofPoposton3.5. Notce that u 0 A so that the admssble set s nonempty. Let {u k } k N A be a mnmzng sequence fo I ε,.e., lm k I ε (u k ) = nf A I ε 0. Hence, u L (; ) C and u L (u 0(); ) C fo some fnte C>0. Applyng a Poncaé nequalty we get that {u k } s bounded n W,, + (; R 2 ). Theefoe, thee s a subsequence convegng weakly* to some u W,, + (; R 2 ). Compactness of the tace opeato ensues that u = u 0 on the bounday of. Consequently,u Aand weak* lowe semcontnuty of I ε fnshes the agument. Indeed, as v s b-quasconvex the weak* lowe semcontnuty of the fst two tems s obvous. The last tem s weak* lowe semcontnuous n vew of Remak Cut-off technque pesevng the b-lpschtz popety One of the man steps n the chaactezaton of gadent Young measues [2, 29] s to show that havng a bounded sequence {y k } k N W, (; R 2 ), such that t conveges weakly to y(x) : R 2 and { y k } geneates a Young measue ν, then thee s a modfed sequence {u k } k N W, (; R 2 ), u k (x) =y(x) fo

14 280 B. BENEŠOVÁ AND M. KRUŽÍK x and { u k } stll geneates ν. Standad poofs of ths fact use a cut-off technque based on convex combnatons nea the bounday; due to the non-convexty of ou constants, howeve, ths could destoy the b-lpschtz popety, so t s not at all sutable fo ou puposes. Theefoe, we esot to a dffeent appoach boowng fom ecent esults by Dane and Patell [3, 4]. Moe pecsely, the followng theoem s a man ngedent of ou appoach. Theoem 5.. Let R 2 be a bounded Lpschtz doman, let dam δ>0 and L be fxed. Then thee exsts ε>0 that s only dependent on δ and L such that f ỹ, y W,, + (; R 2 ) ae L-b-Lpschtz maps satsfyng ỹ y C(;R 2 ) ε(δ, L), then we can fnd anothe c(l)-b-lpschtz map u W,, + (; R 2 ) satsfyng u = y on and {x ; u(x) ỹ(x)} δ. The followng coollay allows us to modfy convegent sequences at the bounday of. Coollay 5.2. Assume that {y k } k N W,, + (; R 2 ) s a sequence of L-b-Lpschtz maps and y k y n W,, + (; R 2 ) as k. Then thee s a subsequence of {y kn } n N and {u kn } n N W,, + (; R 2 ) bounded such that u kn yn W,, + (; R 2 ) as n, fo all n N u kn = y on and lm n {x ; u kn y kn } 0. In patcula, the sequences { y kn } and { u kn } geneate the same Young measue. Poof. Let {δ n } n N be a sequence of postve numbes convegng to zeo as n. We apply Theoem 5. and unfom convegence of {y k } k N to y n C( ; R 2 ) to fnd {ε n (δ n,l)} n N and {y kn } n N such that y kn y C( ;R2 ) ε n (δ n,l). Use Theoem 5. wth ỹ := y kn to obtan u kn W,, + (; R 2 ) wth the mentoned popetes. PoofofTheoem5.. We devote the est of ths secton to povng Theoem 5., lage pats of the poof, collected n ts thd secton, ae athe techncal. Theefoe, we stat wth an ovevew of the poof: Secton of the poof: Ovevew. We defne the open set δ = { x :dst(x, ) <δ } ; now, we fnd = (δ) and a coespondng, sutable (δ)-tlng of δ,.e. a fnte collecton of closed squaes = N D(z,) wth z δ (5.) = that satsfes that δ and that two squaes have n common only ethe a whole edge o a vetex. Futhemoe, we eque the tlng to be fne enough so that thee exsts a collecton of edges Γ satsfyng the followng popetes: evey contnuous path connectng two ponts x and x 2 such that x and x 2 δ \ cosses Γ, Γ nt. Ths settng s best magned n the case when s smply connected. Then, foms a thn stp of squaes nea the bounday and Γ s a closed cuve consstng of edges n the nteo of ths stp. We wll efe to the specal case of a smple connected doman fo a bette magnaton of the ntoduced concepts at seveal places bellow; nevetheless, smple connectvty of s neve explctly used and, n fact, not needed. Futhe, we sepaate nto thee pats: = bulk bound,

15 CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE 28 whee bulk = {x \ : evey contnuous path fom x to cosses.} bound = δ \ ( bulk ). Let us agan, fo a moment, thnk of a smply connected. Then, bulk foms the nteo of the doman, s the thn stp of squaes and bound s also a stp that eaches up to and s not tled. Wth these basc notatons set, we explan how we constuct the cut-off. Let us choose ε = (δ) 2L so that we 3 have that ỹ y C(;R2 ) (δ) 2L 3 (5.2) Now, we alte ỹ on to obtan the functon u δ : R 2 that has the popety that [u δ ] bulk =ỹ and [u δ ] bound = y. If we thnk once moe of smple connected, ths means that on the nne bounday of we obtan the functon ỹ whle on the oute bounday we aleady have the sought bounday data. We wll gve a pecse defnton of u δ n the next secton of the poof. In fact, n vew of the avalable extenson Theoem 2.6, t s suffcent to gve a defnton of u δ on all the edges n, whch we wll explot. Namely, on the edges the fttng of ỹ to y s essentally one-dmensonal and hence ou technque wll be essentally a lnea ntepolaton. In the thd secton of the poof, whch s the most techncal one, we then show that u δ, thus so fa defned only on the edges, s 8L-b-Lpschtz (cf. (5.6)) and so extendng t to va Theoem 2.6 wll yeld a c(l)-b- Lpschtz functon u δ : R 2 havng the above descbed popetes. Indeed, u δ (D(z,)) = u δ ( D(z,)) fo all admssble, sothatu δ : R 2 s njectve. Theefoe, we may defne ỹ(x) on bulk, u(x) = u δ (x) on, y(x) on bound. It s obvous that the obtaned mappng s Lpschtz and satsfes u(x) >c(l) a.e.on. The njectvty of u follows fom the fact that u( bulk ),u( bound )andu( ) ae mutually dsjont, whch s a consequence of the fttng bounday data though [u δ ] bulk =ỹ and [u δ ] bound = y. Thus, the mappng u s globally b-lpschtz and hence oentaton pesevng snce t peseves oentaton on bulk. Secton 2 of the poof: Pattonng of the gd and defnton of u δ. In ths secton we gve a pecse defnton of u δ (x) onthegd of the tlng, denoted Q, whch conssts of all edges of ;nothewods, N Q = D(z,) wth z as n (5.). = Clealy, Γ Qand we dvde Q nto two othe pats defned though Q = Q oute Γ Q nne, Q nne = { x Q\Γ ; evey contnuous path connectng x to cosses Γ }, (5.3) Q oute = Q\(Γ Q nne ). (5.4) The names of these two othe pats ae boowed fom the stuaton when s smply connected; namely, then Q nne coesponds to those edges that ae futhe away fom the bounday than Γ and so n the nteo whle Q oute ae the edges n the exteo. Nevetheless, as aleady stessed above, smple-connectvty of s not needed.

16 282 B. BENEŠOVÁ AND M. KRUŽÍK δ Γ (a) -tlng of the set δ (b) Detal of coss on Γ Fgue. Tlng nea bounday and detal of one coss. Fo futhe convenence, we shall fx some notaton (n accod wth [4]); see also Fgue b. We shall denote w α anyvetexofthegdq that les on Γ, fo any w α we denote wα all vetces that ae at dstance of to w α ; note that fom constucton thee always exst 4 such vetces (as w α cannot le on the bounday of ), fo any w α the lagest numbes ξα > 0thatsatsfy ( ỹ wα + ξα ( )) w α w α y(wα ) = f the edge w α wα Q nne, 4L ( y wα + ξα ( )) w α w α y(wα ) = else; 4L we call the bounday coss the set 4 { Z α = wα + t ( wα w } α) :0 t ξ α = and denote the extemals of ths coss p α...p 4 α. It s due to the L-b-Lpschtz popety of ỹ and y as well as (5.2) that all the concepts above ae well defned. In patcula, we can assue that the numbes ξ α canbefoundnthenteval[/(6l 2 ), /3], (5.5) so that the bounday cosses ae mutually dsjont. We postpone the poof of (5.5) untl the end of ths secton. Now, we ae n the poston to defne the sequence u kδ (x) onq as follows: fst, we defne u δ (x) eveywhee n Q except fo the bounday cosses: {ỹ(x) f x Q nne \ ( α u δ (x) = Z α), y(x) f x (Q oute Γ ) \ ( α Z α);

17 CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE 283 whle on the coss the u δ wll be contnuous and pecewse affne,.e. (ỹ ( ) y(w α )+ t u δ (w α + t(wα w ξ p α α)) = α y(wα ) ) f w α wα Qnne and t [0,ξα ], ( ( ) y(w α )+ t y p α y(wα ) ) f w α wα Q nne and t [0,ξα]. ξ α The ough dea behnd ths constucton s that the matchng, o the cut-off, actually happens on the bounday cosses whee we, on each edge, eplace ỹ as well as y by an affne map. By adjustng the slopes of these affne eplacements we get a contnuous pecewse affne, and hence b-lpschtz, map on the coss. What we need to show ae then, essentally, the followng two popetes of such a eplacement: t connects n a b-lpschtz way to u δ along the endponts of the bounday coss and the adjustment of the slopes needed to obtan contnuty s just small so that the oveall L-b-Lpschtz popety s not affected much. Fo the fome, we mmc the stategy of Dane and Patell [4] who wee also able to connect an affne eplacement of a b-lpschtz functon to the ognal map. The latte s due to the fact that ỹ and y ae sutably close to each othe (as expessed by the popety (5.2)) whch assues that the change of slope on the coss needed fo the cut-off wll depend just on L. We wll show n the next secton that u δ s 8L-b-Lpschtz on Q; cf. (5.6). Theefoe, we can apply Theoem 2.6 to extend u δ fom Q (wthout changng the notaton) to each squae of the tlng. As fo evey squae D(z,) of the tlng we have that u δ (D(z,)) = u δ ( D(z,)) we see that the extended mappng s globally njectve on. Poof of (5.5). Fo w α wα Q nne, we notce that the functon t ỹ( wα + t(wα w α ) ) y(w α ) s contnuous on [0, ] and, owng to (5.2), smalle o equal than 2L n 0 whle n t =wehavethat 3 ỹ(wα ) ỹ(w α)+ỹ(w α ) y(w α ) L 2L 3 4L ; whch yelds the exstence of ξ α [0, ] such that ỹ ( w α + t(w α w α) ) y(w α ) = To establsh the bounds on ξα,wenotethat ỹ 4L = ( wα + ξα (w α w α) ) y(w α ) = ỹ ( w α + ξα (w α w α) ) +ỹ(w α ) ỹ(w α ) y(w α ) Lξα + 2L 3 Lξ α + 2L,.e. ξα /(6L2 ). On the othe hand we have that ỹ 4L = ( wα + ξα (w α w α) ) y(w α ) = ỹ ( w α + ξα (w α w α) ) +ỹ(w α ) ỹ(w α ) y(w α ) ( ξα ) L 2L 2 ( ξα ), L 2 whch s satsfed f 0 ξα /3. In the case when w α wα Q nne, we poceed n a smla way and ely just on the b-lpschtz popety of y; explotng (5.2) s not necessay. Secton 3 of the poof: B-Lpschtz popety of u δ. The functon u δ defned n the pevous secton s contnuous on the gd Q and we clam that t s even b-lpschtz,.e. (as long as (5.2) holds tue) 4L 8L z z u δ (z) u δ (z ) 8L z z z,z Q (5.6)

18 284 B. BENEŠOVÁ AND M. KRUŽÍK The poof of ths clam s the content of ths secton and wll be pefomed n seveal steps. Step of the poof of (5.6): Suppose that z and z le n Z α. Let us fst consde the stuaton when both z,z le on the same edge;.e. z,z w α wα fo some =...4. In ths a case u δ s affne and we have that u δ (z) u δ (z ) u δ (w α ) u δ (p α z z = ) ξα = ξα L ỹ(p α) ỹ(w α)+ỹ(w α) y(w α) y(p α ) y(w α) ξ α L ξ α 2L 3 L 6L2 2L 3 2L f w αwα Qnne 2L f w α w α Qnne. Smlaly, u δ (z) u δ (z ) z z uδ (w α ) u δ (p α ) = = ỹ(p y(p ξ α α ) ỹ(wα)+ỹ(wα) y(wα) ξα L + ξα α ) y(wα) 2L L + 6L2 3 2L 3 2L f w α w α Q nne ξ α L 2L f w α w α Qnne. If z and z ae not on the same edge let, fo example, z w α p α and z w α p 2 α. Moeove, we may assume, wthout loss of genealty, that u δ (z) y(w α ) u δ (z ) y(w α ) and, hence, defne z n the segment w α z such that u δ (z) y(w α ) = u δ (z ) y(w α ). Then, as the ponts u δ (z), u δ (z )andu δ (z ) fom a tangle that s obtuse at u δ (z )(cf. also Fg. 2) wemay apply Remak 5.3 to obtan u δ (z) u δ (z ) ( u δ (z) u δ (z ) + u δ (z ) u δ (z ) ) 2 ( u δ (z) u δ (z ) + ) 2 2L z z (5.7) snce the ponts z, z le on the same edge whee we aleady poved the b-lpschtz popety. Futhe, by the fact that u δ s pecewse affne on the coss 5, ( ) ( ) uδ p α uδ p 2 α u δ (z) u δ (z ) z z = = p α p2 α ỹ(p α) ỹ(p 2 α) p α p2 α L f both p α, p2 α le n Qnne y(p α) y(p 2 α) p α p2 α L f nethe p α no p 2 α les n Q nne ỹ(p α) ỹ(p 2 α)+ỹ(p 2 α) y(p 2 α) p α p2 α L p α p2 α 2L 3 2L, p α Q nne p 2 α,/ Q nne 5 Notce that on any the segment wαp α we can wte u δ (t) =u δ (w α)+t(u δ (p α) u δ (w α)). Theefoe, the ponts z,z coespond to such t, t that t u δ (p α ) u δ(w α) = t u δ (p 2 α ) u δ(w α). By defnton, howeve, u δ (p α ) u δ(w α) = u δ (p 2 α ) u δ(w α) = 4L so that t = t.

19 CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE 285 Fgue 2. The obtuse tangle fomed by u δ (z), u δ (z )andu δ (z ) n the mage of the bounday coss as needed n Step. Notce that snce u δ s pecewse affne on the coss, each segment of the coss foms agan a pat of a staght lne. whee we ealzed that p α p 2 α 6L because the tangle fomed by the ponts p α,w 2 α, p 2 α s ght angled o a lne. Notce also that the stuaton when p α / Qnne, p 2 α Qnne s completely symmetcal to the aleady coveed case. So, etunng to (5.7), we have by the tangle nequalty 2 u δ (z) u δ (z ) 4L z z. On the othe hand, by explotng that the tangle fomed by the ponts z,z and w α s ethe ght angled o a lne, we get that u δ (z) u δ (z ) u δ (z) y(w α )+y(w α ) u δ (z ) 2L ( z w α + z w α ) 2L 2 z z. Step 2 of the poof of (5.6): Suppose that z/ Z α and z / Z β fo all α, β. Notce that we only have to nvestgate the case when z Q nne and z / Q nne fo the othe optons ae tval. Then, howeve, we have that z z and so the Lpschtz popety follows mmedately as 6L 2 u δ (z) u δ (z ) y(z) ỹ(z)+ỹ(z) ỹ(z ) 6L 2 2L + L z z 2L z z. On the othe hand, u δ (z) u δ (z ) z z = y(z) ỹ(z)+ỹ(z) ỹ(z ) z z L 2L 2 z z 2L Step 3 of the poof of (5.6): Suppose that z Z α and z / Z β fo all β. To obtan the lowe bound n (5.6) we ely on Remak 5.4; ndeed the choce of z, z s such that u δ (z ) les outsde the ball B(y(w α ); 4L ) whle u δ(z) B(y(w α ); 4L ). In patcula, we may assume that u δ(z) les on the segment y(w α )u δ (p α ) (ecall that u δ s affne on the coss). So, u δ (z) u δ (z ) u δ(p α ) u δ(z) + u δ (p α ) u δ(z ) 3

20 286 B. BENEŠOVÁ AND M. KRUŽÍK Fgue 3. The obtuse tangle fomed by z,z,p α as needed n Step 2. Clealy, we only have to cae about the latte tem on the ght hand sde. Employng (5.2) and the tangle nequalty, we get that ỹ(p α ) ỹ(z ) p α z L f p α,z Q nne u δ (p α) u δ (z ) p α z y(p α ) y(z ) p α z L f p α,z / Q nne ỹ(p α ) ỹ(z )+ỹ(z ) y(z ) p α z L 6L2 2L 3 2L f p α Q nne and z / Q nne ; whee, n the last case, p α and z necessaly le n dffeent edges and so p α z 6L. Notce that snce the 2 ôle of p α and z s symmetc we eally exhausted all possbltes belongng to ths step. Summng up, u δ (z) u δ (z ) z z 6L To obtan the uppe bound, we fst ealze that f z s at the bounday to the coss,.e. z = p α fo some =,...,4, the pocedue fom Step 2 apples n vebatm. Theefoe, we may estct ou attenton to the stuatonnwhchz s stctly n the nteo of the coss; then, snce all p α ae at dstance at most /3 fomw α and snce z / w α p α, at least one of these p α has to satsfy that the tangle z,p α,z has an obtuse (o ght) angle at p α (see Fg. 3) let t fo notatonal convenence be p α. So, we ae n the poston to apply Remak 5.3 below and estmate u δ (z) u δ (z ) = ( ) ( ) uδ (z) u δ p α + uδ p α uδ (z ) uδ (z) ỹ ( ( ) pα) +ỹ p α ỹ(z ) 2 2L ( z p α + p α z ) 4L z z ) f z Q nne and p α Qnne uδ (z) y ( ) ( ) p α +ỹ p α ỹ(z )+y ( ) ( ) p α ỹ p α 2 2L ( z p α + p α z ) + 6L 2 2L 5L z z f z Q nne and p α / Q nne uδ (z) y ( ( ) pα) + y p α y(z ) 2 2L ( z p α + p α z ) 4L z z ) f z / Q nne and p α / Qnne uδ (z) y ( ) ( ) p α +ỹ p α ỹ(z )+y ( ) ( ) p α ỹ p α 2 2L ( z p α + p α z ) + 6L 2 2L 5L z z f z / Q nne and p α Qnne

21 CHARACTERIZATION OF GRADIENT YOUNG MEASURES GENERATED BY HOMEOMORPHISMS IN THE PLANE 287 whee we used that we aleady poved the b-lpschtz popety nsde the coss Z α andnthesecondand fouth case we used that 6L p 2 α z snce, n ths cases, p α and z have to le on dffeent edges. Step 4 of the poof of (5.6): Suppose that z Z α, z Z β wth α β. The last case we need to consde s when z, z le n two cosses coespondng to two dffeent vetces, espectvely. In such a case w α w β and also, fom defnton, u δ (z ) y(w β ) 4L (as z belongs to the coss). Theefoe, y(w α ) u δ (z ) = y(w β ) y(w α )+u δ (z ) y(w β ) L 4L > 4L ;.e. u δ (z ) / B(y(w α ); 4L ) and we may apply Remak 5.4 to get (wth p α beng the extemal of Z α lyng on the same edge as z) ( ) u δ (z uδ p ) u δ (z) α uδ (z) ( ) + uδ p α uδ (z ) 3 Smlaly, also u δ (p α ) / B(y(w β); 4l )as ( ) y(wβ ) y(w α ) u δ p α + y(wα ) L 4L > 4L ; and hence, agan elyng on Remak 5.4 (p 2 β denotes the extemal of Z β lyng on the same edge as z ) ( ) uδ p α uδ (z) uδ ( ) ( ) ( ) + p α uδ p 2 uδ u δ (z β + p 2 β u δ (z ) ) u δ (z) 9 ( p 8L α z + p α pβ 2 + p 2 β z ) z z 8L, by applyng the tangle nequalty. Moeove, we exploted that u δ (p α ) u δ(z) p α z 2L as p α and z le on the same edge wthn the same coss (cf. Step ); smlaly also fo u δ (p 2 β ) u δ(z ). Fnally, we can see that u δ (p α) u δ (p 2 β ) p α p2 β 2L bythesamepocedueasemployednstep3. It, fnally, emans to pove the uppe bound n (5.6). But ths follows fom the fact that, snce z,z belong to dffeent cosses, thee has to exst a pont p Qthat does not belong to any coss such that the tangle zpz s obtuse (o ght) at p. Hee, we admt also the exteme case n whch zpz le on a staght lne; n ths case, we undestand the angle at p to be π and hence obtuse. Theefoe, explotng (5.3), eadly gves u δ (z) u δ (p)+u δ (p) u δ (z ) 5L ( z p + z p ) 0L ( z z ) 8L z z. 2 Remak 5.3 (Obtuse tangle nequalty). Let us consde a tangle fomed by thee ponts z,p,z R 2 such that the angle γ at p s obtuse o ght (= lage o equal to π/2). Then t follows fom the cosne law z z = z p 2 + z p 2 2 z p z p cos(γ) z p 2 + z p ( z p + z p ). (5.8) Remak 5.4 (Ball sepaaton nequalty). Let us consde a ball centeed at w wth adus ξ and a pont a lyng nsde ths ball on the segment wb wth b w = ξ. Moeove,letc be a pont lyng outsde ths ball. Then, snce b s the neaest to a lyng on the bounday of the mentoned ball t has to hold that a b a c and so by the tangle nequalty 6 a c a b + b c 3 6 Indeed, b c a b + a c 2 a c and so a b + b c 3 a c as desed.