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

We state necessary and sufficient conditions for weak lower semicontinuity of integral functionals of the form ${u\mapsto\int_{\Omega}h(x,u(x))\,\mathrm{d}x}$, where h is continuous and possesses a positively p-homogeneous recession function, ${p>1}$, and ${u\in L^{p}(\Omega;\mathbb{R}^{m})}$ lives in the kernel of a constant-rank first-order differential operator ${\mathcal{A}}$ which admits an extension property. In the special case ${\mathcal{A}=\mathrm{curl}}$, apart from the quasiconvexity of the integrand, the recession function’s quasiconvexity at the boundary in the sense of Ball and Marsden is known to play a crucial role. Our newly defined notions of ${\mathcal{A}}$-quasiconvexity at the boundary, generalize this result. Moreover, we give an equivalent condition for the 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.