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

Abstract

We establish the local $C^{1,\alpha }$ regularity of minimizers for functionals of the form $$w\to {\int }_{\Omega }(|\nabla w{|}^p-fw)dx+{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\frac{|w(x)-w(y){|}^q}{|x-y{|}^{n+sq}}dxdy,$$ where $s\in (0,1)$, $1<p\le sq$, 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.

30 pages