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

¹ Department of Mathematics, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, 09340 Mexico City, Mexico.
² Department of Mathematics, University of Sussex, Pevensey 2 Building, Falmer, Brighton, BN1 9QH, UK.

^* Corresponing author: k.koumatos@sussex.ac.uk

Received: 3 September 2018
Accepted: 2 April 2019

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 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.