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

TL;DR

This paper investigates the regularity properties of almost minimizers for a transmission problem involving the $(p,q)$-Laplacian, establishing universal Hölder and near-Lipschitz continuity under specific conditions.

Contribution

It proves universal Hölder regularity for local almost minimizers and near-Lipschitz regularity when the exponents are close, advancing understanding of regularity in non-standard variational problems.

Findings

Universal Hölder regularity of almost minimizers.

Almost Lipschitz regularity when $|p-q|$ and $ ext{epsilon}$ are small.

Regularity results hold for a broad class of transmission problems.

Abstract

This paper concerns almost minimizers of the functional $$ J(v,\Omega) = \int_\Omega \left( |D v^+|^p + |D v^-|^q \right) dx, $$ where $1<p \neq q< \infty$ 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 $\epsilon\ll 1$.