Regularity of vectorial minimizers for non-uniformly elliptic anisotropic integrals

We establish the local boundedness of the local minimizers $u:\mathrm{\Omega }\to {\mathbb{R}}^m$ of non-uniformly elliptic integrals of the form ${\int }_{\mathrm{\Omega }}f(x,Dv)dx$, where $\mathrm{\Omega }$ is a bounded open subset of ${\mathbb{R}}^n$ ($n\ge 2)$ and the integrand satisfies anisotropic growth conditions of the type $$\sum _{i=1}^n{\lambda }_i(x)\mathrm{\mid }{\xi }_i{\mathrm{\mid }}^{p_i}\le f(x,\xi )\le \mu (x)\{1+\mathrm{\mid }\xi {\mathrm{\mid }}^q\}$$ 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 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.