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

This paper concerns almost minimizers of the functional $$J(v,\Omega) = \int_\Omega \left( |D v^+|^p + |D v^-|^q \right) dx,$$ where $1<p \neq q< \infty$ and $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\geq 1$. We prove the universal H\"older 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 $\epsilon\ll 1$.