Null Lagrangian Measures in subspaces, compensated compactness and conservation laws

TL;DR

This paper characterizes when Null Lagrangian Measures are trivial on subspaces of matrices, linking their existence to rank-1 connections, and applies these results to conservation laws in elasticity.

Contribution

It provides a necessary and sufficient condition for triviality of Null Lagrangian Measures on linear subspaces, resolving an open problem and extending understanding of measure support in PDE analysis.

Findings

Null Lagrangian Measures are trivial iff the subspace has no rank-1 connections for dimensions 1-3.

Non-trivial Null Lagrangian Measures exist on subspaces with rank-1 connections in dimensions 1-3.

Results apply to entropy solutions of 2x2 conservation law systems in elasticity.

Abstract

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 $\mathcal{M}^{pc}(\mathcal{K})$ was an open question even in the case where $\mathcal{K}$ is a linear subspace of $M^{m\times n}$. We answer this question and provide a necessary and sufficient condition for any linear subspace $\mathcal{K}\subset M^{m\times n}$. The ideas also allow us to show that for any $d\in \left\{1,2,3\right\}$, $d$-dimensional subspaces $\mathcal{K}\subset M^{m\times n}$ support non-trivial Null Lagrangian Measures if and only if $\mathcal{K}$ has Rank-$1$ connections. This is known to be false for $d\ge 4$. 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.