On Trivial Gradient Young Measures

We give a condition on a closed set K of real n mma trices which ensures that any W p gradient Young measure sup ported on K must be trivial the condition given is also necessary when K is bounded Introduction Assume is a smooth bounded domain in R and p is a given number Let W p R be the usual Sobolev space of maps u from to R the Jacobi or gradient matrix ru of u is L integrable and thus de ned point wise by ru i u i x i n m The notion of strong convergence and weak convergence inW p R is de ned as usual and denoted by and respectively Let M M m be the space of all real n m real matrices A with standard norm jAj tr A A For any subset K of M let dK be the distance function to K i e dK A inf P K jP Aj A M Following Kinderlehrer and Pedregal see also Ball a family of probability measures x x on M is said to be a W p gradient Young measure if there exists a weakly convergent sequence fujg in the Sobolev space W p R such that i for all C M the sequence f ruj g converges weakly in L to function x R M A d x A ii fjrujj g converges weakly in L loc M In this case instead of frujg we say fujg is a determining sequence of the gradient Young meausre x x Notice that condition ii is required in the de nition and that the determining sequence fujg may not be unique We refer to for Mathematics Subject Classi cation J Q A BAISHENG YAN a characterization of all such W 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 di erential equa tions see e g Lemma A W p gradient Young measure x x is a Dirac mea sure for almost every x if and only if every determining sequence fujg of x x converges strongly in W loc R n In this case we say x x is trivial Although nontrivial Young measures play an important role in many physical problems see and the references therein it is the trivial Young measures that are mostly related to certain compactness prob lems encountered in the study of nonlinear partial di erential equa tions see e g In this paper we study the W p gradient Young measures x x supported on a given closed subset K that is supp x K for almost every x We give some conditions on the supporting set K which ensure that any gradient Young measures supported on K be trivial The main results In the rest of the paper we assume K is a closed subset of M Suppose x x is a W p gradient Young measures supported on K that is supp x K a e x If x x is trivial then it is immediate that Z M Ad x A ru x K a e x where u W p R is a map uniquely up to an additive constant determined by the determining sequences of x x In general it is di cult if not impossible to verify condition A condition on set K which ensures any W p gradient Young measure x x supported on K satisfy has been given in Yan One of the main results of this paper is the following Theorem Let p Assume condition holds for all W p gradient Young measures x x supported on K Suppose also that for ON TRIVIAL GRADIENT YOUNG MEASURES each there exists C such that for some cube Q in R m Z