On weak and viscosity solutions of nonlocal double phase equations

TL;DR

This paper studies nonlocal double phase equations, proving local Hölder continuity of solutions and exploring the relationship between weak and viscosity solutions under certain conditions.

Contribution

It introduces new regularity results for solutions and clarifies the connection between weak and viscosity solutions for nonlocal double phase equations.

Findings

Weak solutions are locally Hölder continuous.

Established equivalence between weak and viscosity solutions.

Extended De Giorgi-Nash-Moser methods to nonlocal double phase context.

Abstract

We consider the nonlocal double phase equation \begin{align*} \mathrm{P.V.} &\int_{\mathbb{R}^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{sp}(x,y)\,dy\\ &+\mathrm{P.V.} \int_{\mathbb{R}^n} a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{tq}(x,y)\,dy=0, \end{align*} where $1<p\leq q$ and the modulating coefficient $a(\cdot,\cdot)\geq0$. Under some suitable hypotheses, we first use the De Giorgi-Nash-Moser methods to derive the local H\"{o}lder continuity for bounded weak solutions, and then establish the relationship between weak solutions and viscosity solutions to such equations.