Interior $C^{1,\alpha}$ regularity of mixed local-nonlocal $(p,q)$-energy minimizers for $p\leq sq$

TL;DR

This paper proves that minimizers of a mixed local-nonlocal energy functional are locally $C^{1, eta}$ regular under certain conditions, extending previous results to cover all relevant parameter ranges.

Contribution

It establishes the local $C^{1, eta}$ regularity of minimizers for a class of mixed local-nonlocal functionals, completing the regularity theory for all $p, q, s$ with bounded sources.

Findings

Proves local $C^{1, eta}$ regularity of minimizers.

Completes the regularity theory for all $p, q, s$ ranges.

Extends previous work by covering all parameter cases.

Abstract

We establish the local $C^{1, \alpha}$ regularity of minimizers for functionals of the form $$w\to \int_{\Omega}(|\nabla w|^p-fw) dx + \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|w(x)-w(y)|^q}{|x-y|^{n+sq}}dx\, dy,$$ where $s \in (0, 1)$, $1 < p \leq sq$, and $f \in L^\infty(\Omega)$. This result complements the work of De Filippis and Minigione in \cite{DFM}, thereby completing the proof of $C^{1,\alpha}$ regularity for all $p, q \in (1, \infty)$ and $s \in (0, 1)$ with locally bounded source term.