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 C 1 extremals defined on this class of non-smooth domains. The sufficiency result presented here thus extends the seminal theorem by Grabovsky & Mengesha (2009), where smoothness assumptions are made on the boundary.
We consider functionals of the form $$\begin{equation*} \mathcal{F}(u):=\int_\Omega\!F(x,u,\nabla u)\,\mathrm{d} x, \end{equation*}$$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded. The integrand $F\colon\Omega\times\mathbb{R}^N\times\mathbb{R}^{N\times n}\to\mathbb{R}$ is assumed to satisfy the classical assumptions of a power p-growth and the corresponding strong quasiconvexity. In addition, F is Hölder continuous with exponent $2\beta\in(0,1)$ in its first two variables uniformly with respect to the third variable and bounded below by a quasiconvex function depending only on $z\in\mathbb{R}^{N\times n}$. We establish that strong local minimizers of $\mathcal{F}$ are of class $\operatorname{C}^{1,\beta}$ in an open subset $\Omega_0\subseteq\Omega$ with $\mathcal{L}^n(\Omega\setminus\Omega_0)=0$. This partial regularity also holds for a certain class of weak local minimizers at which the second variation is strongly positive and satisfying a bounded mean oscillation (BMO) smallness condition. This extends the partial regularity result for local minimizers by Kristensen and Taheri (2003) to the case where the integrand depends also on u. Furthermore, we provide a direct strategy for this result, in contrast to the blow-up argument used for the case of homogeneous integrands.
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