Regularity of vectorial minimizers for non-uniformly elliptic anisotropic integrals

Abstract

We establish the local boundedness of the local minimizers $u:\hat{I}\text{c◯}\to {\mathbb{R}}^m$ of non-uniformly elliptic integrals of the form ${\int }_{\hat{I}}\text{c◯}f(x,Dv)dx$, where $\hat{I}\text{c◯}$ is a bounded open subset of ${\mathbb{R}}^n$ ($n\ge 2)$ and the integrand satisfies anisotropic growth conditions of the type \[ \sum_{i=1}^{n}λ_{i}(x)|ξ_{i}|^{p_{i}}\le f(x,ξ)\leμ(x)\left\{ 1+|ξ|^{q}\right\} \] for some exponents $q\ge p_i>1$ and with non-negative functions $\hat{I}{»}_i,\hat{I}¼$ fulfilling suitable summability assumptions. The main novelties here are the degenerate and anisotropic behaviour of the integrand and the fact that we also address the case of vectorial minimizers ($m>1$). Our proof is based on the celebrated Moser iteration technique and employs an embedding result for anisotropic Sobolev spaces.