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

TL;DR

This paper introduces boundary quasiconvexity conditions for non-smooth domains, establishing their necessity and sufficiency for strong local minimality of ${ m C}^1$ extremals in the vectorial Calculus of Variations, extending prior smooth boundary results.

Contribution

It extends the quasiconvexity-based sufficiency theorem to non-smooth domains, relaxing boundary smoothness assumptions in the calculus of variations.

Findings

Necessary quasiconvexity conditions for non-smooth domains identified

A sufficiency theorem for ${ m C}^1$ extremals established

Extension of classical results to non-smooth boundary 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.