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1 Max-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig A sharp version of hang's theorem on truncating sequences of gradients by Stefan Muller Preprint-Nr.:

3 A sharp version of hang's theorem on truncating sequences of gradients Stefan Muller Max Planck Institute for Mathematics in the Sciences Inselstr Leipzig, Germany Abstract Let K mn be a compact and convex set of m n matrices and let fujg be a sequence in W 1 1 loc (n m ) that converges to K in the mean, i.e. n dist(duj K)! 0. I show that there exists a sequence vj of Lipschitz functions such that kdist(dvj K)k 1! 0 and L n (fuj 6= vjg)! 0. This renes a result of Kewei hang (Ann. S.N.S. Pisa 19 (1992), ) who showed that one may assume kdvjk 1 C. Applications to gradient Young measures and to a question of Kinderlehrer and Pedregal (Arch. at. Mech. Anal. 115 (1991), 329{365) regarding the approximation of [f+1g valued quasiconvex functions by nite ones are indicated. Achallenging open problem is whether convexity ofk can be replaced by quasiconvexity. Mathematical eview Classication: 49J45 1 Main results Let fu j g be a sequence of weakly dierentiable functions u j : n! m whose gradients approach the ball B(0 ) in the mean, i.e. dist(du j B(0 ))dx! 0: (1.1) n Motivated by work of Acerbi and Fusco [AF 84], [AF 88], and Liu [Li 77] Kewei hang showed that the sequence can be modied on a small set in such a way that the new sequence is uniformly Lipschitz. The following Theorem is a slight variant oflemma3.1in[h92]. 1

4 Theorem 1 (hang). There exists a constant c(n m) with the following property. If (1.1) holds then there exists a sequence of functions v j : n! m such that jjdv j jj 1 c(n m) L n (fu j 6= v j g)! 0: In fact one has the seemingly stronger conclusions L n (fu j 6= v j or Du j 6= Dv j g)! 0 jdu j ; Dv j jdx! 0: n For the rst conclusion it suces to note that for weakly dierentiable functions u and v the implication u = v a:e: in A =) Du = Dv a:e: in A (1.2) holds (see e.g. [GT 83], Lemma 7.7). For the second conclusion observe that jdu j ; Dv j jjdv j j + jdu j jc(n m) + + dist(du j B(0 )) and integrate over the set fdu j 6= Dv j g. Theorem 1 has found important applications to the calculus of variations, in particular the study of quasiconvexity, lower semicontinuity, relaxation and gradient Young measures ([KP 91], [h 92], see also Corollary 3). The purpose of the present work is to show that the constant c(n m) can be chosen arbitrarily close to 1 and that the ball B(0 ) can be replaced by a compact, convex set. Theorem 2 Let K be a compact, convex set in mn. Suppose that u j 2 W 1 1 loc (n m ) and n dist(du j K)dx! 0: (1.3) Then there exists a sequence v j of Lipschitz functions such that emarks. jjdist(dv j K)jj 1! 0 L n fu j 6= v j g! 0: 1. A more natural and apparently much harder question is whether the same assertion holds if K is quasiconvex rather than convex. 2

5 2. Jan Kristensen pointed out to me that in the scalar case m = 1 the assumption that K be convex can be dropped. Let CK denote the convex hull of K and Cdist K the convex envelope of the distance function. Kristensen's proof uses (2.14), applied with CK and (CK) = CK, the identity Cdist K =dist (CK) and the relaxation of nonconvex integral functionals (see e.g. [ET 76]) to obtain inff dist(dv K )dy : v = u B = inff B Cdist(Dw K ):w = u B dist(d~u (CK) ) (1 ; 3 ;n ) B dist(du CK)dx: A similar argument can be applied for m > 1 provided that a (somewhat articial) condition holds which is slightly stronger than the requirement that CK agrees with the quasiconvex hull QK of K. In the language of Young measures (see [KP 91], [KP 94] for the relevant denitions) one can deduce the following. Corollary 3 Let K be as a compact, convex set in mn and let n be open, let p 1, and suppose that fu j g generates a W 1 p gradient Young measure = f x g x2 and that supp x K for a:e: xin : Then there exists a sequence fv j g that generates the same gradient Young measure and satises jjdist(dv j K)jj 1! 0: Warning: There are slightly dierent denitions of W 1 1 gradient Young measures in use. Above we have adopted the convention that those measures are generated by sequences for which fdu j g is equiintegrable (and not merely bounded in L 1 ). No ambiguities arise for p>1. Using Corollary 3 one can simplify the theory of W 1 1 gradient Young measures and answer some of the questions raised in [KP 91], see Corollary 9below. 3

6 A version of Corollary 3 for Young measures with nite pth moment was discovered by Kristensen [Kr 94] and later independently in [FMP 97]. It can be used to obtain a simpler approach to W 1 p gradient Young measures ([Pe 97], [Sy 97]). For 6= n Corollary 3 requires a local version of Theorem 2. Theorem 4 Let n beopen and let fu j g be asequence inw 1 1 loc ( m ) that satises u j! u 0 in L 1 loc( m ) (1.4) dist(du j K)! 0 in L 1 loc(): (1.5) Then there exists an increasing sequence ofopen sets U j,compactly contained in, and functions v j 2 W 1 1 loc ( m ) such that emarks. v j = u 0 on n U j (1.6) L n (fu j 6= v j g\u j )! 0 (1.7) jjdist(dv j K)jj 1! 0: (1.8) 1. If has nite volume we have L n ( n U j )! 0 and thus L n (fu j 6= v j g)! If u j *u 0 in W 1 1 ( loc m ) then (1.4) holds by the compact Sobolev embedding. In fact (1.4) and (1.5) imply weak convergence in W 1 1 ( loc m ) (see proof). 3. Condition (1.6) is a statement of the fact that u 0 and v j satisfy the same `boundary condition' (traces may not exist since we assumed no regularity of). 2 Proofs in n In order to remove the small `bad' region where dist(du j K) > we locally mollify u j. Akey point is to use dierent mollication radii in dierent regions of n (I learned about the use of x-dependent molliers through the papers [SU 82] and [SU 83] of Schoen and Uhlenbeck). Each mollication step reduces the L 1 norm of the distance function by a xed factor but 4

7 slightly increases the L 1 norm on the good set. however, that the latter eect can be controlled. Careful iteration shows, A more precise outline of the proof is as follows. In Lemma 5 we obtain quantitative estimates for mollication on a ball. In Lemma 6 we combine these estimates with a covering argument to achieve the desired reduction of the L 1 norm. Theorem 7 contains the result of the iteration procedure. Finally Theorem 2 is an immediate consequence of Theorem 7. In the following K always denotes a compact, convex set in M := M mn = mn. We use the operator norm jf j := supfjfxj : jxj = 1g on M. The distance function dist(a K) = minfja ; F j : F 2 Kg is 1-Lipschitz and convex, since K is convex. Its sublevel sets K := fa M : dist(a K) g are compact and convex, and for > 0 >0onehas (K ) = K + dist(a K ) (dist(a K) ; ) + (2.1) where a + = max(a 0). If we let we have By E jkj 1 := maxfjaj : A 2 Kg (2.2) jk j 1 = jkj 1 + jaj jkj 1 +dist(a K): (2.3) fdx we denote the mean value (L n (E)) ;1 fdx. E Lemma 5 If u 2 W 1 1 (B(a r) m ) and if 1 jkj 1 dist(du K)dx B(a r) < 8 ;(n+1) : =9 1 n+1 jkj1 then there exists ~u 2 W 1 1 (B(a r)) such that ~u = u r) 5

8 B(a r) dist(d~u K )dx (1+ 1 n+1 ) B(a r)nb(a r=2) dist(du K)dx: Proof. The statement is invariant under the rescaling u! r u( x ; a ) K! K! : jkj 1 r jkj 1 jkj 1 Wemaythus assume jk 1 j =1 a=0 r=1,andwewrite B := B(0 1) B := B(0 ). Let 2 (0 1=8) (a specic choice will be made below), and for x 2 B 7=8 let Then v(x) = B(x ) udy= Dv(x) = B u(x + z)dz: B(x ) and, by convexity of the distance function, dist(dv(x) K) B(x ) Du dy dist(du K)dy ;n : Let ' : B! [0 1] be a cut-o function that satises and dene Then ~u = u on B n B 7=8 and ' 2 W (B 7=8 ) ' 1 on B 5=8 jd'j 8 ~u =(1; ')u + 'v: D~u =(1; ')Du + 'Dv +(v ; u) D' in B: Thus D~u = Du in B n B 7=8 dist(d~u K) = dist(dv K) ;n in B 5=8 : (2.4) 6

9 We next estimate v ; u. In view of (2.3) and the assumption jk 1 j = 1 we have for a.e. x 2 B 7=8 jv ; uj(x) ju(y) ; u(x)jdy B(x ) = 1 L n (B ) 1 L n (B ) + 1 L n (B ) We have T 1 (x ) = n n+1 1 L n (B ) 0 0 S n;1 ju(x + e) ; u(x)jdh n;1 (e) n;1 d 0 S n;1 0 0 S n;1 0 =: T 1 (x )+T 2 (x ): 1dt dh n;1 (e) n;1 d dist(du K)(x + te)dt dh n;1 (e) n;1 d and thus Fubini's theorem yields B 7=8 nb 5=8 (jv ; uj;)dx S n;1 0 0 BnB 1=2 B 7=8 nb 5=8 1 C T 2 (x )dx dist(du K)dxA dt dh n;1 (e) n;1 d (2.5) BnB 1=2 (distdu K)dx: Since the distance function is convex and 1-Lipschitz we have dist(d~u K) ' dist(du K)+(1; ')dist(dv K)+jv ; ujjd'j Let dist(du K)+ ;n +8 +8(jv ; uj;): = 1 n+1 : 7

10 Then ;n +8 = and dist(d~u K) dist(du K)+ +8(jv ; uj;) in B 7=8 n B 5=8 : The estimate (2.4) gives dist(d~u K) < in B 5=8 : Since D~u = Du in B n B 7=8 the assertion follows from (2.5), (2.1) and the denition of. 2 Lemma 6 There exist positive constants (n) < 1 c 2 (n) < 1=8, with the following property. If u 2 W 1 1 loc (n m ), 2 (0 9c 2 jkj 1 ) and if := 1 dist(du K) < 1 jkj 1 m then there exists a function ~u 2 W 1 1 loc (n m ) such that 1 dist(du K ) (n) (2.6) jkj 1 n emark. L n (fu 6= ~ug) 2 n 9jKj1 If Du 2 K on n n V then n+1 : (2.7) fu 6= ~ug V = fx : dist(x V ) g = c 7 (jkj n+1 1 ;(n+1) ) 1=n : Proof. 1. We may suppose jkj 1 =1. Let E := ( 9 )n+1 <c (n+1) 2 < 8 ;(n+1) := fx 2 n :sup r dist(du K)dy > g: B(x r) 8

11 Note that by the Lebesgue point theorem dist(du K) a:e: in n n E: (2.8) Since c 2 1 we have ( 9 )n : (2.9) 2. Claim: For each x 2 E there exists a radius (x) > 0 such that dist(du K)dy B(x (x)) dist(du K)dy =: (2.10) B(x (x)=2) To prove the claim consider the function and let h(r) := dist(du K)dy B(x r) (x) =2supfr 2 (0 1) :h(r) g: Then (x) < 1 since < 1 and h((x)=2) = by continuity of h. The claim is proved. 3. For (x) as above consider the family of closed balls F = fb(x (x)) : x 2 E g: By the Besicovitch covering theorem there exist at most k(n) (countable) subfamilies F (j) of disjoint balls such that the unionofthesets A (j) = [ B2F (j)b covers E. Thus there exists a subfamily F 0 of disjoint balls such that the set A = [ B2F 0B satises A dist(du K)dy 1 dist(du K)dy: (2.11) k(n) E 9

12 4. In view of (2.10) we may apply Lemma 5 successively to each of the disjoint balls B(x i i ) 2 F 0 to obtain a function ~u 2 W 1 1 loc (n m ) that satises ~u = u in n n A (2.12) dist(d~u K )dy (1 + 1 n+1 ) dist(du K)dy: B(x i i ) B(x i i )nb(x i i =2) The denition of i = (x i ) (see (2.10)) implies that dist(du K)dy 2 ;n dist(du K)dy: (2.13) B(x i i =2) B(x i i ) Hence (2.13) yields dist(d~u K )dy (1 ; 2 ;n )(1+ 1 n+1 ) dist(du K)dy: B(x i i ) B(x i i ) Let c 2 =min(c 2 1=9) where c 2 is dened by the equation (2.14) (1 ; 2 ;n )(1 + c 2 )=(1; 3 ;n ): Then the denition of implies that (1 ; 2 ;n )(1 + 1 n+1 ) (1 ; 3 ;n ): Since the balls in F 0 are disjoint and their union is A we deduce that A dist(d~u K )dy (1 ; 3 ;n ) A dist(du K)dy: (2.15) On the other hand (2.12) yields in combination with (2.8) and (2.9) dist(d~u K ) = dist(du K ) in n n A Thus dist(du K ) = 0 in n n E : dist(d~u K )dy dist(du K)dy: n na E na 10

13 If we add this to (2.15), use (2.11) and dene (n) =1; 3;n k(n) (2.16) we nally obtain dist(d~u K )dy (n) dist(du K)dy: n E This proves the rst assertion of the lemma. 5. To nish the proof it only remains to estimate L n (A). One has P L n (A) = L n (B(x i i )) B(x i i )2F 0 =2 n P L n (B(x i i =2)) =2 n P 1 B(x i i =2) 2n =2n ( 9 )n+1 : dist(du K)dy Hence (2.7) holds and the lemma is proved. 2 Proof of the remark. The function u is only modied on the balls B(x i i ) and one has (see point 5. above) L n (B(x i i )) 2 n 9 n+1 ;(n+1) 2 i c 7 ( ;(n+1) ) 1=n = : Now wemust have x i 2 V i V =2 since otherweise B(x i i dist(du K)dx = ) 0. Thus B(x i i ) V. 2 Let c 2 = c 2 (n) be as in Lemma 6. Theorem 7 There exists a constant c(n) with the following property. Suppose that K is a compact, convex set in mn, u 2 W 1 1 loc (n m ), 2 11

14 (0 9c 2 jkj 1 ) and := 1 jkj 1 n dist(du K)dx < 1: Then there exist v 2 W 1 1 ( n m ) such that Dv 2 K a:e: L n fu 6= vg c(n)jkj n+1 1 ;(n+1) : Corollary 8 If Du 2 K on n n V then fu 6= vg V = c 9 (jkj n+1 1 ;(n+1) ) 1=n : In fact values of u outside V play no r^ole in the construction of v. Proof of Theorem 7. By scaling we may suppose jkj 1 = 1. The proof is based on a simple iteration of Lemma 6. Let = (n) denote the constant in (2.16) and inductively dene K 0 = K K i+1 =(K i ) i M i = jk i j 1 i = i 2(n+1) Mi : The value of >0willbechosen below. We have ln M i+1 M i =ln M i + i M i i 2n+1 M0 =1 and hence 1 M i e c 3 =: M 1 X i=0 i c 4 c c 3 =: : Construct a sequence u i by successive application of Lemma 6, starting with u 0 = u. Let i = 1 M i n dist(du i K i )dy i = L n fu i+1 6= u i g: 12

15 By Lemma 6 i+1 i Thus i i 2 n 9 n+1 M n+1 ;(n+1) i i : i i c 5 M n+1 ;(n+1) i=2 where c 3 c 4 and c 5 only depend on the space dimension n. Since P i < 1 it follows from the denition of i and (1.2) that Moreover P u i! v Du i! h in measure: n dist(du i K ) Mi! 0: (2.17) Application of the dominated convergence theorem with majorant jkj + dist(du i K ) shows that Du i! h in L 1 loc (n M), and testing with i smooth, compactly supported test functions we deduce u i! v W 1 1 loc (n m ): Moreover, by (2.17), and Now choose such that L n (fu 6= vg) Dv 2 K a:e: 1X i=0 i c 6 M n+1 i ;(n+1) : = = c 4 e c 3 : (2.18) Since 9c 2 we have 9c 2 c ;1 4, c ;1 4 exp(9c 2 c 3 c ;1 4 ) and the choice c(n) c 6 c n+1 4 exp(18(n +1) c 2c 3 c 4 ) gives the desired estimate for L n (fu 6= vg). 2 13

16 Proof of Corollary 8. Let u i be as in the proof of Theorem 7andlet V i = V [fu i 6= ug i = c 7 ( M n+1 ;(n+1) i i ) 1=n : We have Du i 2 K in n n V i, and the remark after Lemma 6 yields V i+1 (V i ) i : Since i i the denition of i implies that X n+1 i c 8M n ; n+1 1 n n : The assertion follows from (2.18). 2 3 Local estimates Proof of Theorem 4. We may suppose jkj 1 =1. 1. Claim: u j *u 0 W 1 1 loc ( m ) Du 0 2 K a.e. Proof. Let U be open, U (as usual this notion indicates that U is compact and contained in ). For A 2 mn let PA denote the best approximation of A in the convex, compact set K. The sequence PDu j is bounded in L 1 (U) and hence there exists a subsequence that has a weak limit h in L 1 (U). Since U is bounded, in particular Now PDu jk *hin L 1 (U): jdu j ; PDu j j =dist(du j K)! 0inL 1 (U) and hence Du jk * h in L 1 (U). The usual argument yields h = Du 0, and uniqueness of the limit implies that the whole sequence converges. Convexity of the distance function and Mazur's and Fatou's lemma (or standard lower semicontinuity results) show that dist(du 0 K)=0 a.e. in U, and hence a.e. in by arbitrariness of U. 14

17 2. Let V U. We construct v j that almost satisfy (1.7) and (1.8). The proof will then be nished by a diagonalization argument. Let ' 2 C 1 0 (V ) 0 ' 1, and dene w j = 'u j +(1; ')u 0 : Then Dw j = 'Du j +(1; ')Du 0 +(u j ; u 0 ) D': In particular Dw j 2 K in n V j := dist(dw j K)dx dist(du j K)dx + ju j ; u 0 j jd'j dx: V By the assumptions j! 0. Let > 0. In view of Theorem 6 and Corollary 8 there exist j 0 = j 0 (U V ' ) such that for all j j 0 there exist v j 2 W 1 1 ( loc m ) that satisfy It follows that fv j 6= w j gu L n (v j 6= w j ) < Dv j 2 K a.e. v j = u 0 in n U L n (fv j 6= u j g\u) <+ L n (f' 6= 1g\V )+L n (U n V ) dist(dv j K) : 3. Let f Uk ~ g be an increasing sequence of open sets Uk ~ whose union exhausts. Let V k U ~ k ' k 2 C 1 0 (V k ) such that V L n ( ~ Uk n V k ) < 1 k Ln (f' k 6=1g\V ) < 1 k 0 ' k 1 and let k < 1 k. By point 2. there exists j k such that for j j k there exist functions v j that satisfy v j = u 0 in n ~ Uk L n (fv j 6= u j g) \ ~ Uk ) < 3 k dist(dv j K) < 1 k : 15

18 We may suppose without loss of generality thatj k is (strictly) increasing. To nish the proof dene U j = ~ Uk if j k j <j k +1: 2 4 Application to quasiconvex functions A function f from the m n matrices mn to [f;1 1g is called quasiconvex if for all bounded domains U n with L n (@U) =0andallF 2 mn U f(f + D)dx U f(f )dx = L n (U)f(F ) 8 2 W (U m ) whenever the integral on the left exists. Quasiconvexity is the fundamental notion in the vector-valued calculus of variations (see [Mo 52], [Mo 66], [Ba 77], [Da 89], [Ev 90], [Sv 94]). It states that ane functions minimize the functional u 7! f(du) subject to their U own boundary conditions. Quasiconvexity is dicult to handle, however, since no local characterization is known for n m > 1 (and cannot exist for m 3 n 2, see [Kr 97]). Even the approximation of general quasiconvex functions by nite ones is a largely open question. As a corollary of Theorem 2we obtain at least the following result that answers the question in [KP 91], p. 350, equation (5.19) (see pp. 342 and 345 for the relevant denitions). We remark that every -valued quasiconvex function is continuous and even locally Lipschitz since it is rank-1 convex (see e.g. [Da 89]). Corollary 9 Let K mn be a convex, compact set with non-empty interior. Let f : mn! [f;1 1g be a quasiconvex function that satises Then, for all F 2 K, f 2 C(K ) f =+1 on mn n K: f(f ) = supfg(f )jg : mn! g f on K g quasiconvexg: (4.1) Proof. 16

19 1. We may assume 0 2 int K, since quasiconvexity is invariant under translation in mn. We have K int K 8 >1: (4.2) Indeed, if A then ta +(1; t)b 2 K for all t 2 (0 1) and all B in a small neighbourhood of 0. Hence ta 2 int K, for all t 2 (0 1). Thus (4.2) holds. 2. Let G 1 denote the right hand side of (4.1) and let P denote the nearest neighbour projection onto K. For k 2 N [f0g dene h k (F )=f(pa)+k dist(a K) f: Let g k = h qc k denote the quasiconvex hull of h k, i.e. the largest quasiconvex function below h k. Thus g k (F ) G 1. On the other hand by standard relaxation results (see e.g. [Da 89], Chapter 5, Theorem 1.1) g k (F ) = inff Q h k (Du)dx : u ; Fx 2 W (Q m )g where Q =(0 1) n. Hence there exist Lipschitz functions u k such that In particular lim sup k!1 Q h k (Du k )dx G 1 u k = Fx (4.3) Q dist(du k K)! 0: Hence Du k is bounded in L 1 and, after possible passage to a subsequence we may assume that u k! u 0 in L By Theorem 4 there exist v k 2 W 1 1 (Q m ) which satisfy L n (fu k 6= v k g)! 0 v k = Fx (4.4) jjdist(dv k K)jj 1! 0: (4.5) 17

20 Taking into account (1.2), the uniform continuity ofh 0 and the inequality h 0 h k we see that lim sup k!1 Q h 0 (Dv k )dx = lim sup k!1 Q h 0 (Du k )dx G 1 : In view of (4.2) and (4.5) there exist k & 1 such that ;1 k Dv k 2 K, ;1 k F 2 K. Using the uniform continuityofh 0 as well as quasiconvexity and continuity of f we obtain f(f )= lim f( ;1 k!1 k = lim sup k!1 Q F ) lim sup k!1 Q h 0 ( ;1 k Dv k)dx G 1 : f( ;1 k Dv k)dx The proof is nished. 2 Acknowledgements It is a pleasure to thank G. Alberti, J. Kristensen and V. Sverak for very interesting discussions. 18

21 eferences [AF 84] [AF 88] [Ba 77] [Da 89] [ET 76] [Ev 90] [FMP 97] [GT 83] [KP 91] [KP 94] [Kr 94] E. Acerbi and N. Fusco, Semincontinuity problems in the calculus of variations, Arch. at. Mech. Anal. 86 (1984), 125{145. E. Acerbi and N. Fusco, An approximation Lemma for W 1 p functions, in: Material instabilities in Continuum mechanics and related mathematical problems, (J.M. Ball, ed.) Oxford UP, J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. at. Mech. Anal. 63 (1977), B. Dacorogna, Direct methods in the calculus of variations, Springer, I. Ekeland and. Temam, Convex analysis and variational problems, North Holland, Amsterdam, L.C. Evans, Weak convergence methods for nonlinear partial dierential equations, CBMS no. 74, 1990, Am. Math. Soc. I. Fonseca, S. Muller and P. Pedregal, Analysis of concentration and oscillation eects generated by gradients, to appear in SIAM J. Math. Anal. D. Gilbarg and N.S. Trudinger, Elliptic partial dierential equations of second order, Springer, 2nd. ed., D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients, Arch. at. Mech. Anal. 115 (1991), 329{365. D. Kinderlehrer and P.Pedregal, GradientYoung measure generated by sequences in Sobolev spaces, J. Geom. Analysis 4 (1994), 59{90. J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions, Ph.D. Thesis, Technical University of Denmark, Lyngby. 19

22 [Kr 97] [Li 77] [Mo 52] [Mo 66] [Pe 97] [Sy 97] [SU 82] [SU 83] [Sv 94] [h 92] J. Kristensen, On the non-locality of quasiconvexity, to appear in Ann. IHP Anal. non lineaire. F.-C. Liu, A Luzin type property of Sobolev functions, Ind. Univ. Math. J. 26 (1977), 645{651. C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacic J. Math. 2 (1952), 25{53. C.B. Morrey, Multiple integrals in the calculus of variations, Springer, P. Pedregal, Parametrized measures and variational principles, Birkhauser, M. Sychev, A new approach to Young measure theory, relaxation and convergence in energy, manuscript.. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Dier. Geom. 17 (1982), 307{335.. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Dier. Geom. 18 (1983), 253{268. V. Sverak, Lower semicontinuity of variational integrals and compensated compactness, in: Proc. ICM 1994, vol. 2 Birkhauser, 1995, pp. 1153{1158. K. hang, A construction of quasiconvex functions with linear growth at innity, Ann. S.N.S. Pisa 19 (1992), 313{

2 BAISHENG YAN When L =,it is easily seen that the set K = coincides with the set of conformal matrices, that is, K = = fr j 0 R 2 SO(n)g: Weakly L-qu

2 BAISHENG YAN When L =,it is easily seen that the set K = coincides with the set of conformal matrices, that is, K = = fr j 0 R 2 SO(n)g: Weakly L-qu RELAXATION AND ATTAINMENT RESULTS FOR AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL BAISHENG YAN Abstract. Consider functional I(u) = R jjdujn ; L det Duj dx whose energy-well consists of matrices