ON TRIVIAL GRAIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -gradient Young measure supported on K must be trivial the condition given is also necessary when K is bounded. 1. Introduction Assume is a smooth bounded domain in R m and 1 p<1 is a given number. Let W 1 p ( R n ) be the usual Sobolev space of maps u from to R n thejacobi or gradient matrix ru of u is (L p integrable and thus) dened point-wise by (ru) i = @u i =@x 1 i n 1 m: The notion of strong convergence and weak convergence in W 1 p ( R n ) is dened as usual and denoted by \!" and\*", respectively. Let M = M nm be the space of all real n m real matrices A with standard norm jaj = (tr(a t A)) 1=2 : For any subset K of M, let d K be the distance function to K i.e., d K (A) = inf jp ; Aj P 2K (A 2 M): Following Kinderlehrer and Pedregal [6] (see also Ball [1]), a family of probability measures ( x ) x2 on M is said to be a W 1 p -gradient Young measure if there exists a weakly convergent sequence fu j g in the Sobolev space W 1 p ( R n ) such that (i) for all 2 0 (M) the sequence f(ru j )g converges weakly * in L 1 () to function (x) R = M (A) d x(a) (ii) fjru j j p g converges weakly in L 1 loc ( M): In this case, instead of fru j g we say fu j g is a determining sequence of the gradient Young meausre ( x ) x2 : Notice that condition (ii) is required in the denition and that the determining sequence fu j g may not be unique. We refer to [6] for 1991 Mathematics Subject lassication. 49J45, 4920, 35A15. 1
2 BAISHENG YAN a characterization of all such W 1 p -gradient Young measures and for further information. The following result links the Young measures to the compactness property of certain approximate solutions in partial dierential equations see, e.g., [1, 3, 6, 8, 9] Lemma 1. A W 1 p -gradient Young measure ( x ) x2 is a irac measure for almost every x 2 if and only if every determining sequence fu j g of ( x ) x2 converges strongly in W 1 1 ( loc Rn ): In this case, we say ( x ) x2 is trivial. Although nontrivial Young measures play an important role in many physical problems (see [2, 8] and the references therein), it is the trivial Young measures that are mostly related to certain compactness problems encountered in the study of nonlinear partial dierential equations see, e.g., [3, 9]. In this paper, we study the W 1 p -gradient Young measures ( x ) x2 supported on a given closed subset K that is, supp x Kfor almost every x 2 : We give some conditions on the supporting set K which ensure that any gradient Young measures supported on K be trivial. 2. The main results In the rest of the paper, we assume K is a closed subset of M: Suppose ( x ) x2 is a W 1 p -gradient Young measures supported on K that is, supp x K a:e: x 2 : If ( x ) x2 is trivial, then it is immediate that (1) M Ad x(a) =ru(x) 2K a:e: x 2 where u 2 W 1 p ( R n ) is a map uniquely (up to an additive constant) determined by the determining sequences of ( x ) x2. In general, it is dicult, if not impossible, to verify condition (1). A condition on set K which ensures any W 1 p -gradient Young measure ( x ) x2 supported on K satisfy (1) has been given in Yan [10]. One of the main results of this paper is the following. Theorem 2. Let 1 <p<1: Assume condition (1) holds for all W 1 p - gradient Young measures ( x ) x2 supported onk: Suppose also that for
ON TRIVIAL GRAIENT YOUNG MEASURES 3 each >0 there exists > 0 such that for some cube in R m (2) jrj jrj p +1 + d p K (A + r) for all A 2 K and 2 W 1 p 0 ( R n ): Then any W 1 p -gradient Young measure supported on K must be trivial. We remark that condition (2) does not depend on the particular cube the proof of this result is left to the interested reader. Lemma 3. If condition (2) holds for one cube then it holds for all cubes parallel with moreover, the constants in (2) are also independent of such cubes. When K is compact, condition (2) turns out also necessary for any W 1 p -gradient Young measure supported on K to be trivial. We shall also prove the following. Theorem 4. Let KM be compact and 1 <p<1: Assume condition (1) holds for all W 1 p -gradient Young measures ( x ) x2 supported on K: Then any W 1 p -gradient Young measure supported onk is trivial if and only if condition (2) is valid. To verify the condition (2), we give a sucient condition which is motivated by the work of Sverak [8] see also [7]. A function F : M M! R is said to separate K from gradients provided that,foracube in R m (3) F (X Y ) > 0 F (X X + r) 0 for all X Y 2K X 6= Y and 2 1 0 ( R n ): Note that this condition is independent of : The following result can be considered as a generalization of the result on incompatible two-well problems in [7, 8]. Proposition 5. Let K M be a compact set. Suppose F (X Y ) is continuous, separates K from gradients and satises (4) jf (X Y )j (1 + jy j p ) jr Y F (X Y )j (1 + jy j p;1 ) for all X 2K Y 2 M: Then condition (2) holds for K:
4 BAISHENG YAN Proof. For any given A 2Kand 2 1 0 for all x 2 ( R n ) let K(x) 2Ksatisfy (5) d K (A + r(x)) = ja + r(x) ; K(x)j: Let B(x) = A + r(x) ; K(x): Then B + K ; A = r and jbj = d K (A + r): Therefore, to prove (2), it suces to prove jb(x)+k(x) ; Aj dx jj + jb(x)j p dx which is equivalent to (6) jk(x) ; Aj dx jj + jb(x)j p dx: We prove (6) by deriving some point-wise estimates. Since K is compact, thus from (3), it follows that jx ; Y j=2+ F (X Y ) 8 X Y 2K for all >0 where 1 is a constant depending only on : Therefore, (7) jk ; Aj =2+ F (A B + K) ; [F (A B + K) ; F (A K)]: On the other hand, by the growth condition (4), we deduce jf (A B + K) ; F (A K)j (1 + jbj p;1 ) jbj Since B + K = A + r(x) we have by (7) =2+ jbj p : jk(x) ; Aj + F (A A + r(x)) + jb(x)j p and hence (6) follows by integrating the both sides of this inequality over and using (3). The proof is completed. 3. Proof of the main results In this section, we prove our main results: Theorems 2and4. We need a well-known unford-pettis compactness theorem see, e.g., [4, page 27-II]. Lemma 6. Let R m bealebesgue measurable set and S abounded set in L 1 (): Then for every sequence in S to contain a subsequence
ON TRIVIAL GRAIENT YOUNG MEASURES 5 which converges weakly in L 1 () it is necessary and sucient that for any >0 there exists a >0 such that (8) sup v2s E jv(x)j dx < 8 E jej <: Proof of Theorem 2. To prove the theorem, we assume ( x ) x2 is any W 1 p -gradient Young measure supported on K and fu j g is a determining sequence of ( x ) x2. By Lemma 1, it is sucient toshow that fu j g converges strongly to u in W 1 1 ( 0 ) for all 0 where u is the weak limit of fu j g in W 1 p ( R n ): First of all, by (1), we have ru(x) 2Kfor a.e. x 2 : We followthe idea of Evans and Gariepy [5] to proceed. Without loss of generality, we assume that the sides of cube are parallel to the axes of R m : Let ; 0 be the set of all closed cubes having side length 1 and vertices lying in m of integer lattices. For k = 1 2 inductively dene ; k to be the set of cubes obtained by cutting each cube in ; k;1 into 2 km identical cubes with side length 2 ;k. We extend ru by zero outside and dene M k (x) by M k (x) = X 2;k (x) jj ru(z) dz where is the characteristic function of : Then, M k! ru in L p () and thus g(m k )! g(ru) in L 1 () for all continuous functions g satisfying 0 g(x) (jxj p +1): Let > 0 and 1 < 1 be given as in (2) and let 0 be given. Select k large enough so that every cube in ; k that intersects 0 must be compactly contained in and also (9) km k ;ruk Lp () + kd K (M k )k Lp () <: Let 1 2 N denote the cubes in ; k which intersect 0 and thus are compactly contained in : Let 00 = [ N l : Then 0 00 : For 0 < < 1 let l be the concentric and parallel cubes inside the cube l with side length =2 k for each l =1 2 ::: N: For each l =1 2 ::: N let l 2 1 0 ( l ) be cut-o functions satisfying 0 l 1 l l =1 jr l j2 k (1 ; ) ;1 :
6 BAISHENG YAN Let l j l (u j ; u): hoose K l 2Kso that d K (M k (x)) = ja l ; K l j for x 2 l where A l = M k (x) = 1 ru for x 2 l : j l j l We now compute (10) (11) 00 d p K (ru j) = NX NX l d p K (ru j) l d p K (Kl + r l j) + E 1 + E 2 where E 1 = E 2 = NX NX l n l l d p K (ru j) ; d p K (ru + rl j) d p K (ru + rl j) ; d p K (Kl + r l j) : Note that, by assumption (2) and Lemma 3, where NX E 0 = l d p K (Kl + r l j) NX l Hence, by (11), we have (12) NX NX jr l jj p +1 E 3 = l jr l jj;e 0 l jru j ;ruj;e 0 0 jru j ;ruj;e 3 ; E 0 NX l n l jru j ;ruj: 0 00 jru j ;ruj d pk (ru j) ; E 1 ; E 2 + E 3 + E 0 : We need to estimate E 0 E 1 E 2 and E 3 :
ON TRIVIAL GRAIENT YOUNG MEASURES 7 We rst estimate E 0 E 1 and E 3 as follows. p;1 je 3 j M meas([ N ( l n )) l p NX je 0 j 1+jru j j p + jruj p + jr l j p ju j ; uj p je 1 j l M + M (1 ; ) ;p ku j ; uk p L p () NX 1+jru j j p + jruj p + jr l j p ju j ; uj p l n l meas([ N ( l n l )) + M (1 ; ) ;p ku j ; uk p L p () where M is independent of j and and (s) is a function such that (s)! 0 as s! 0 which follows from the unford-pettis theorem (Lemma 6) and the assumption that fjru j j p g converges weakly in L 1 (): In order to estimate E 2 we write E 2 = E 21 + E 22 where NX E 21 = d p K (ru + rl j) ; d p K (Al + r l j) E 22 = We also note that NX l l d p K (Al + r l j) ; d p K (Kl + r l j) : (13) jd p K (X) ; dp K (Y )j K p (jxj p;1 +1)jX ; Y j 8 X Y 2 M thus we can estimate that je 21 j je 22 j NX l kru ; M k k Lp () NX NX l l kd K (M k )k Lp () 1+jruj p;1 + jm k j p;1 + jr ljj p;1 jru ; M k j M +(1; ) 1;p ku j ; uk p;1 L p () 1+jM k j p;1 + jk l j p;1 + jr ljj p;1 jk l ; M k j 1+2jM k j p;1 + d p;1 K (M k)+jr l jj p;1 d K (M k ) M +(1; ) 1;p ku j ; uk p;1 L p () :
8 BAISHENG YAN Therefore, by (9) (14) (15) je 2 j (je 21 j + je 22 j) 1+(1; ) 1;p ku j ; uk p;1 : L p () Let j!1in (12) by (15) we have lim sup j!1 0 jru j ;ruj + meas([ N ( l n l )) Letting! 1 ; since meas([ N (l n ) l! 0and(0 + )=0 we thus have lim sup j!1 0 jru j ;ruj where < 1 is independent of > 0: Since > 0 is arbitrary, it follows that lim sup jru j ;ruj =0 j!1 0 which shows that fu j g converges strongly to u in W 1 1 ( 0 R n ) for all 0 and hence, by Lemma 1, the Young measure ( x ) x2 is trivial. The proof of Theorem 2 is now complete. Proof of Theorem 4. Let K be a compact set and assume that any W 1 p -gradient Young measures ( x ) x2 supported on K must be trivial. We show condition (2) is valid. Without loss of generality, : We use the contradiction method. Suppose condition (2) were not valid then there would exist 0 > 0 A j 2 K and j 2 W 1 p 0 ( R n ) such that, for each j =1 2 :::, (16) jr j j > 0 (jr j j p +1)+j d p K (A j + r j ): We extend j by zero onto and denote the new functions also by j : Let u j (x) = A j x + j (x) then u j 2 W 1 p ( R n ) and ru j (x) 2 K for a.e. x 2 n : Since K is bounded, it is easy to see that there exist positive constants c 1 c 2 such that d p K (A j + B) c 1 jbj p ; c 2 8 B 2 M j =1 2 ::: From this and (16) it is quite easy to see that fjr j j p g is a bounded sequence in L 1 (). Hence, since 1 <p<1 there exists a subsequence, :
ON TRIVIAL GRAIENT YOUNG MEASURES 9 denoted the same, j *weakly in W 1 p 0 ( R n ): It is also easily seen that u j *u A 0 x + for some A 0 2Kand, by (16), that (17) lim j!1 d p K (ru j)=0: Let ( x ) x2 be a family of probability measures on M also called a generalized Young measure, such that condition (i) in the denition of W 1 p -gradient Young measures given in the introduction holds we refer to [1, 9] for the existence and other properties of such generalized Young measures. From (17) we also have supp x Kfor a.e. x 2 : We now show that ( x ) x2 is a W 1 p -gradient Young measure. By denition, this will be proved if we show that the sequence fjru j j p g contains a subsequence which converges weakly in L 1 () or, equivalently,ifwe show that fjr j j p g contains a subsequence whichconverges weakly in L 1 (): From the denition of j we deduce E jr j j p c 0 1 jej + c 0 2 c 0 1 jej + c0 2 j E\ d p K (A j + r j ) jr j j and, consequently, for any >0we can nd a >0 such that sup j=1 2 ::: E jr j j p < 8 E jej <: By Lemma 6, this shows that fjr j j p g contains a subsequence which converges weakly in L 1 (): We have proved that ( x ) x2 is a W 1 p -gradient Young measure supported on K: By assumption, ( x ) x2 is trivial then Lemma 1 im- that u j! u and thus j! in W 1 1 ( R n ) hence, by (16), jrj 0: Rplies On the other hand, the triviality of( x ) x2 implies ru(x) =A 0 + r(x) 2K a:e: x 2 : Therefore, u is Lipschitz map and uj @ = A 0 x: We can then use this map u to construct a W 1 p -gradient Young measure ( x ) x2 supported on K with a determining sequence fv j g weakly * converges to A 0 x in W 1 1 ( R n )andsatisfying moreover F (rv j (x)) dx = F (ru(x)) dx
10 BAISHENG YAN for all continuous functions F : M! R: Such a construction uses the self-similar structures and a well-known technique of Vitali's covering see, e.g., [6, 10]. Since this Young measure ( x ) x2 must be also trivial that is, v j! A 0 x in W 1 1 ( 0 R n ) for all 0 using F (A) =ja ; A 0 j above, we deduce that r 0 which contradicts R jrj 0 > 0: The proof of Theorem 4 is now complete. References 1. J. M. Ball, A version of the fundamental theorem for Young measures, in \Partial ierential Equations and ontinuum Models of Phase Transitions," (M. Rascle et al. eds.), Lecture Notes in Physics, No. 344, Springer, New York, 1988. 2. J. M. Ball and R.. James, Proposed experimental tests of a theory of ne microstructures and the two well problem, Phil. Trans. Roy. Soc. London, 338A (1992), 389{450. 3. R. J. iperna, ompensated compactness and general systems of conservation laws, Trans. A.M.S., 292(2) (1985), 383{420. 4.. ellacherie and P.-A. Meyer, \Probabilities and Potential," North-Holland, 1978. 5. L.. Evans and R. F. Gariepy, Some remarks concerning quasiconvexity and strong convergence, Proc. Roy. Soc. Edinburgh, 106A (1987), 53{61. 6.. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal., 4(1) (1994), 59{90. 7. J. Matos, Young measures and the absence of ne microstructures in a class of phase transitions, Euro. J. Appl. Math., 3 (1992), 31{54. 8. V. Sverak, On the problem of two wells, in \Microstructure and Phase Transition," 183{190, (. Kinderlehrer et al. eds.), Springer-Verlag, Berlin, 1993. 9. L. Tartar, The compensated compactness method applied to systems of conservation laws, in \Systems of Nonlinear Partial ierential Equations," (J. M. Ball ed.), NATO ASI Series, Vol. III,. Reidel, 1983. 10. B. Yan, Relaxation of certain partial dierential relations in Sobolev spaces, Preprint epartment of Mathematics, Michigan State University, East Lansing, MI 48824, USA E-mail address: yan@math.msu.edu