Boundary regularity and sufficient conditions for strong local minimizers

TL;DR

This paper provides a new proof for the sufficiency of strong local minimizers in the calculus of variations, focusing on boundary regularity and conditions ensuring minimality in the quasiconvex setting.

Contribution

It introduces a novel proof approach for sufficiency theorems, including boundary regularity and regularity results for extremals with VMO gradients.

Findings

Partial boundary regularity for strong local minimizers

Full regularity for Lipschitz extremals with VMO gradients

A new decomposition theorem for variations

Abstract

In this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning $C^1$-extremals at which the second variation is strictly positive. The results are presented in the quasiconvex setting, in accordance with the original statement by Grabovsky and Mengesha (2009). The strategy that we follow relies on a decomposition theorem that allows to split a sequence of variations into its oscillating and its concentrating parts, as well as on a sufficiency result according to which smooth extremals are spatially-local minimizers. Furthermore, we prove partial regularity up to the boundary for strong local minimizers in the non-homogeneous case and a full regularity result for Lipschitz extremals with gradient of vanishing mean oscillation. As a consequence, we also establish a sufficiency result for this class of extremals. The regularity results are established via a blow-up argument.