Null Lagrangian Measures in Subspaces, Compensated Compactness and Conservation Laws

Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set K. Mmxn such that lim j.8 dist(Du j, K) L p = 0 and sup j u j W1, p < 8. {Du j} j is precompact in L p. (1) Let M1, M2,..., Mq denote the set of all minors of Mmxn. A sufficient condition for (1) is that any probability measure mu supported on K satisfying Mk (X) d mu(X) = Mk Xd mu(X) for all k (2) is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [ 21], we denote the set of Null Lagrangian Measures supported on K byMpc(K). For general m, n, a necessary and sufficient condition for triviality ofMpc(K) was an open question even in the case where K is a linear subspace of Mmxn. We answer this question and provide a necessary and sufficient condition for any linear subspace K. Mmxn. The ideas also allow us to show that for any d. {1, 2, 3}, d- dimensional subspaces K. Mmxn support non- trivial Null Lagrangian Measures if and only if K has Rank- 1 connections. This is known to be false for d = 4 from [ 5]. Further using the ideas developed we are able to answer a question of Kirchheim et al. [ 18]. Let P1(u, v) :=.. u v a(v) u ua(v) 1 2 u2 + F(v).. and K1 := {P1(u, v) : u, v. IR} for some function a and its primitive F. The set K1 arises in the study of entropy solutions to the 2 x 2 system of conservation laws ut = a(v) x and vt = ux. In [ 18], the authors asked what are the conditions on the function a such that Mpc(K1 n U) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in K1. Given a = (a1, a2). IR2, if a (a2) > 0 then we construct non-trivial measures inMpc(K1 n Bd (P1(a))) for anyd > 0. On the other hand if a (a2) < 0 then for sufficiently smalld > 0, we show thatMpc(K1 n Bd (P1(a))) consists of Dirac measures.