Local behaviour of the mixed local and nonlocal problems with nonstandard growth

TL;DR

This paper investigates the local behavior of minimizers in mixed local and nonlocal variational problems with nonstandard growth, establishing regularity results such as boundedness, Hölder continuity, and Harnack inequalities.

Contribution

It extends existing regularity results to cases where the local and nonlocal exponents satisfy p ≤ sq, broadening the understanding of such mixed problems.

Findings

Proves local boundedness of minimizers.

Establishes local Hölder continuity.

Derives Harnack inequality for minimizers.

Abstract

We consider the mixed local and nonlocal functionals with nonstandard growth \begin{eqnarray*} u\mapsto\int_{\Omega}(|Du|^p-f(x)u)\,dx+\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^q}{|x-y|^{N+sq}}\,dxdy \end{eqnarray*} with $1<p\le sq$, $0<s<1$ and $\Omega\subset\mathbb{R}^N$ being a bounded domain. We study, by means of expansion of positivity, local behaviour of the minimizers of such problems, involving local boundedness, local H\"{o}lder continuity and Harnack inequality. The results above can be seen as a natural extension of the results under the center condition that $p>sq$ in [De Filippis-Mingione, Math. Ann., https://doi.org/10.1007/s00208-022-02512-7].