$\mathcal{A}$-quasiconvexity and weak lower semicontinuity of integral functionals

Abstract

We state necessary and sufficient conditions for weak lower semicontinuity of $u\mapsto\int_\Omega h(x,u(x))\,d x$ where $|h(x,s)|\le C(1+|s|^p)$ is continuous and possesses a recession function, and $u\in L^p(\Omega;\mathbb{R}^m)$, $p>1$, lives in the kernel of a constant-rank first-order differential operator $\mathcal{A}$ which admits an extension property. Our newly defined notion coincides for $\mathcal{A}=\operatorname{curl}$ with quasiconvexity at the boundary due to J.M. Ball and J. Marsden. Moreover, we give an equivalent condition for weak lower semicontinuity of the above functional along sequences weakly converging in $L^p(\Omega;\mathbb{R}^m)$ and approaching the kernel of $\mathcal{A}$ even if $\mathcal{A}$ does not have the extension property.