H\"older regularity results for parabolic nonlocal double phase problems

TL;DR

This paper proves higher H"older regularity for solutions to nonlocal parabolic double phase problems involving fractional operators, extending regularity results to time-dependent and stationary cases with specific coefficient conditions.

Contribution

It establishes new higher H"older regularity results for weak solutions to nonlocal double phase problems, including both time-dependent and stationary cases, under certain coefficient conditions.

Findings

Higher space-time H"older continuity for time-dependent problems

Global H"older continuity for stationary problems

Regularity results under locally continuous coefficients

Abstract

In this article, we obtain higher H\"older regularity results for weak solutions to nonlocal problems driven by the fractional double phase operator \begin{align*} \mc L u(x):=&2 \; {\rm P.V.} \int_{\mathbb R^N} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps_1}}dy \nonumber &+2 \; {\rm P.V.} \int_{\mathbb R^N} a(x,y) \frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{N+qs_2}}dy, \end{align*} where $1<p\leq q<\infty$, $0<s_2, s_1<1$ and the modulating coefficient $a(\cdot,\cdot)$ is a non-negative bounded function. Specifically, we prove higher space-time H\"older continuity result for weak solutions of time depending nonlocal double phase problems for a particular subclass of the modulating coefficients. Using suitable approximation arguments, we further establish higher (global) H\"older continuity results for weak solutions to the stationary problems involving the operator $\mc L$ with modulating coefficients that are locally continuous.