Some remarks on the higher regularity of minimizers of anisotropic functionals

We consider the anisotropic integral functional of the calculus ofvariations ∫Ω [∑ᵢ₌₁ⁿ cᵢ |Dᵢu|pᵢ] dx, where Ω⊂ Rⁿ (n≥ 2), is an open boundedset, u:Ω→ 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:λ |t|pᵢ ≤ fᵢ(t) ≤ Λ |t|pᵢfor every i=1,, n and some positive constants0<λ<Λ. Moreover, some further assumption, such as a{ strong ellipticity} condition of the kind∑ᵢ₌₁ⁿ f''ᵢ(ξᵢ) ηᵢηᵢ ≥ ν ∑ᵢ₌₁ⁿ|ξᵢ|pᵢ-2 |ηᵢ|², is assumed for everyξ,η∈ Rⁿ and some ν >0.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 Ω.