Regularity of vectorial minimizers for non-uniformly elliptic anisotropic integrals

We establish the local boundedness of the local minimizers $u:\Omega \to R^m$ of non-uniformly elliptic integrals of the form ${\int }_{\Omega }f(x,Dv)dx$, where $\Omega$ is a bounded open subset of $R^n$ $(n\ge 2)$ and the integrand satisfies anisotropic growth conditions of the type $\sum _{i=1}^n{\lambda }_i\left(x\right){\left|{\xi }_i\right|}^{p_i}\le f(x,\xi )\le \mu \left(x\right)\left(1+{\left|\xi \right|}^q\right)$for some exponents $q\ge p_i>1$ and with non-negative functions ${\lambda }_i,\mu$ fulfilling suitable summability assumptions. The main novelties here are the degenerate and anisotropic behavior 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.