Boundary regularity and sufficient conditions for strong local minimizers

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.

arXiv commentRevised version, now in press, Journal of Functional Analysis, 2017

References in corpus (2)

• Direct approach to the problem of strong local minima in Calculus of Variations
• Necessary and sufficient conditions for the strong local minimality of $C^1$ extremals on a class of non-smooth domains

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