Higher regularity for minimizers of very degenerate integral functionals

Abstract

In this article, we consider minimizers of integral functionals of the typeF(u)=∫Ωa(x)p(|Du|−1)+pdx with a bounded domain Ω⊂Rn(n≥2), a growth exponent p≥2 and Lipschitz continuous coefficients a:Ω→R. We consider the vectorial setting, i.e. u:Ω→RN with N≥1. We will prove that H(Du) is continuous for any continuous function H:RNn→R vanishing on {ξ∈RNn:|ξ|≤1}. This extends a recent result from [3] to the case of integrands that explicitly depend on the x-variable.