Null Lagrangian Measures in planes, compensated compactness and conservation laws

Andrew Lorent, Guanying Peng

Published 2018-01-09Version 1

Compensated compactness is an important method used to solve nonlinear PDEs. A simple formulation of a compensated compactness problem is to ask for conditions on a set $\mathcal{K}\subset M^{m\times n}$ such that $$ \lim_{n\rightarrow \infty} \mathrm{dist}(Du_n,\mathcal{K})\overset{L^p}{\rightarrow} 0\; \Rightarrow \{Du_{n}\}_{n}\text{ is precompact.} $$ Let $M_1,M_2,\dots, M_q$ denote the set of minors of $M^{m\times n}$. A sufficient condition for this is that any measure $\mu$ supported on $\mathcal{K}$ satisfying $$ \int M_k(X) d\mu (X)=M_k\left(\int X d\mu (X)\right)\text{ for }k=1,2,\dots, q $$ is a Dirac measure. We call measures that satisfy the above equation "Null Lagrangian Measures" and we denote the set of Null Lagrangian Measures supported on $\mathcal{K}$ by $\mathcal{M}^{pc}(\mathcal{K})$. For general $m,n$, a necessary and sufficient condition for triviality of $\mathcal{M}^{pc}(\mathcal{K})$ is unknown even in the case where $\mathcal{K}$ is a linear subspace of $M^{m\times n}$. We provide a condition that is sufficient for any linear subspace $\mathcal{K}$ and also necessary in the case where $2\leq\min\left\{m,n\right\}\leq 3$. A corollary is that two dimensional subspaces $\mathcal{K}\subset M^{m\times n}$ support nontrivial Null Lagrangian Measures if and only if $\mathcal{K}$ has Rank-$1$ connections. Using the ideas developed we are able to answer (up to first order) a question of Kirchheim, M\"{u}ller and Sverak on the Null Lagrangian measures arising in the study of a (one) entropy solution of a $2\times 2$ system of conservation laws that arises in elasticity. Further the methods lead to a strategy to provide a more direct proof of DiPerna's well known result on existence of entropy solutions to the system. The strategy could potentially be applied to other systems.