Some remarks on the higher regularity of minimizers of anisotropic functionals

We consider the anisotropic integral functional of the calculus ofvariations _ ₈=₁ⁿ cᵢ |Dᵢu|^pᵢ dx, where Rⁿ (n 2), is an open boundedset, u: R, cᵢ 0 for i=1, , n are constants and fᵢ: R [0, +) are functions satisfying, forevery t R the following non standard growth condition: |t|^pᵢ fᵢ (t) |t|^pᵢ for every i=1, , n and some positive constants00. The aim of the article is to exhibit a minimizer of such functional, for an opportune choice of the exponents pᵢ, which turns out to be bounded everywhere and Lipschitz continuous (or even of class C¹) in a opportune subset of.