Necessary and sufficient conditions for the strong local minimality of C1 extremals on a class of non-smooth domains

Motivated by applications in materials science, a set of quasiconvexity at the boundary conditions is introduced for domains that are locally diffeomorphic to cones. These conditions are shown to be necessary for strong local minimisers in the vectorial Calculus of Variations and a quasiconvexity-based sufficiency theorem is established for C1 extremals defined on this class of non-smooth domains. The sufficiency result presented here thus extends the seminal theorem by Grabovsky and Mengesha (2009), where smoothness assumptions are made on the boundary.

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.