Almost minimizers to a transmission problem for $(p,q)$-Laplacian

Abstract

This paper concerns almost minimizers of the functional $$J(v,\Omega )={\int }_{\Omega }\left(|D{v}^{+}{|}^p+|D{v}^{-}{|}^q\right)dx,$$ where $1<p\ne q<\infty$ and $\Omega$ is a bounded domain of ${\mathbb{R}}^n$, $n\ge 1$. We prove the universal Hölder regularity of local $(1+\epsilon )$-minimizers, when $\epsilon$ is universally small. Moreover, we prove almost Lipschitz regularity of the local $(1+\epsilon )$-minimizers, when $|p-q|\ll 1$ and $\epsilon \ll 1$.

19 pages; cross-reference fixed