# Hölder regularity for nonlocal double phase equations

@article{DeFilippis2019HlderRF,
title={H{\"o}lder regularity for nonlocal double phase equations},
author={Cristiana De Filippis and Giampiero Palatucci},
journal={Journal of Differential Equations},
year={2019}
}
• Published 17 January 2019
• Mathematics
• Journal of Differential Equations
We prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient $a=a(\cdot,\cdot)$. The model case is driven by the following nonlocal double phase operator, $$\int \!\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\, {\rm d}y + \int \!a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y… Local boundedness and Hölder continuity for the parabolic fractional p-Laplace equations In this paper, we study the boundedness and Hölder continuity of local weak solutions to the following nonhomogeneous equation$$\begin{aligned} \partial _tu(x,t)+\mathrm{P.V.}\int _{{\mathbb
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We prove $C^{1,\nu}$ regularity for local minimizers of the \oh{multi-phase} energy: \begin{flalign*} w \mapsto \int_{\Omega}\snr{Dw}^{p}+a(x)\snr{Dw}^{q}+b(x)\snr{Dw}^{s} \ dx, \end{flalign*} under
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