Null Lagrangian Measures in subspaces, compensated compactness and conservation laws

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 $$\underset{n\to \mathrm{\infty }}{\lim }\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(D{u}_n,\mathcal{K})0\Rightarrow \{D{u}_n{\}}_n\text{ is precompact.}$$ 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 $$\int M_k(X)d\mu (X)=M_k(\int Xd\mu (X))\text{ for }k=1,2,\dots ,q$$ 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 $d\in \{1,2,3\}$, $d$-dimensional subspaces $$\mathcal{K}$\subset M^{m\times n}$supportnon-trivialNullLagrangianMeasuresifandonlyif$$\mathcal{K}$$hasRank-$1$connections\mathrm{.}Thisisknowntobefalsefor$d\ge 4$. 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.