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

This paper concerns almost minimizers of the functional $$J(v,\mathrm{\Omega })={\int }_{\mathrm{\Omega }}(\mathrm{\mid }D{v}^{+}{\mathrm{\mid }}^p+\mathrm{\mid }D{v}^{-}{\mathrm{\mid }}^q)dx,$$ where $1<p\mathrm{\ne }q<\mathrm{\infty }$ and $\mathrm{\Omega }$ 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 $\mathrm{\mid }p-q\mathrm{\mid }\ll 1$ and $\epsilon \ll 1$.