Higher regularity for minimizers of very degenerate convex integrals

Abstract

In this paper, we consider minimizers of integral functionals of the type F(u)≔∫Ω1p(|Du(x)|γ(x)−1)+pdx,for p>1, where u:Ω⊂Rn→RN, with N≥1, is a possibly vector-valued function. Here, |⋅|γ is the associated norm of a bounded, symmetric and coercive bilinear form on RnN. We prove that K(x,Du) is continuous in Ω, for any continuous function K:Ω×RnN→R vanishing on {(x,ξ)∈Ω×RnN:|ξ|γ(x)≤1}.