Hölder regularity results for parabolic nonlocal double phase problems

Abstract

In this article, we obtain higher Hölder 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 $1<p\le q<\infty$, $0<s_2,s_1<1$ and the modulating coefficient $a(\cdot ,\cdot )$ is a non-negative bounded function. Specifically, we prove higher space-time Hölder 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) Hölder continuity results for weak solutions to the stationary problems involving the operator $\mc L$ with modulating coefficients that are locally continuous.

To appear in Adv. Diff. Equ