Hölder regularity results for parabolic nonlocal double phase problems (2112.04287v4)

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.