Necessary and sufficient conditions for the strong local minimality of $C^1$ extremals on a class of non-smooth domains

Abstract

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 ${\rm C}^1$ 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.