H\"older regularity for weak solutions to nonlocal double phase problems

TL;DR

This paper establishes local boundedness and H"older continuity for weak solutions to nonlocal double phase problems, identifying sharp conditions on coefficients and exponents that mirror local double phase regularity results.

Contribution

It provides the first regularity results for nonlocal double phase problems, extending local double phase theory to fractional nonlocal operators.

Findings

Proved local boundedness of solutions.

Established H"older continuity under sharp conditions.

Identified conditions on coefficients and exponents for regularity.

Abstract

We prove local boundedness and H\"older continuity for weak solutions to nonlocal double phase problems concerning the following fractional energy functional \[ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}} + a(x,y)\frac{|v(x)-v(y)|^q}{|x-y|^{n+tq}}\, dxdy, \] where $0<s\le t<1<p \leq q<\infty$ and $a(\cdot,\cdot) \geq 0$. For such regularity results, we identify sharp assumptions on the modulating coefficient $a(\cdot,\cdot)$ and the powers $s,t,p,q$ which are analogous to those for local double phase problems.