Accepted Paper

Inserted: 25 mar 2025
Last Updated: 24 jun 2025

Year: 2025
Doi: https://doi.org/10.48550/arXiv.2503.18917

Abstract:

We establish the local boundedness of the local minimizers $u:\Omega\rightarrow\mathbb{R}^{m}$ of non-uniformly elliptic integrals of the form $\int_{\Omega}f(x,Dv)\,dx$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$ ($n\geq2)$ and the integrand satisfies anisotropic growth conditions of the type \[ \sum_{i=1}^{n}\lambda_{i}(x)
\xi_{i}
^{p_{i}}\le f(x,\xi)\le\mu(x)\left\{ 1+
\xi
^{q}\right\} \] for some exponents $q\geq p_{i}>1$ 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.

Keywords: anisotropic Sobolev spaces, local boundedness, Degenerate anisotropic growth, p, q-growth conditions