Boundary regularity and sufficient conditions for strong local minimizers

In this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning C1-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 [31]. 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.