On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data

TL;DR

This paper proves the equivalence of weak and viscosity solutions for nonlocal double phase problems with nonhomogeneous data, and establishes local boundedness of weak solutions under certain conditions.

Contribution

It demonstrates the equivalence between weak and viscosity solutions for a class of nonlocal double phase problems with nonhomogeneous data, extending existing theory.

Findings

Equivalence between weak and viscosity solutions established.

Local boundedness of weak solutions proved in specific cases.

Results applicable to nonlocal operators with nonhomogeneous data.

Abstract

This work focuses on the nonhomogeneous nonlocal double phase problem \begin{align*} L_au(x)=f(x,u,D_s^p u, D_{a,t}^q u) \text{ in } \Omega, \end{align*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $0<s,t<1<p\leq q<\infty$ with $tq\leq sp$ and the operator $L_a$ is defined as \begin{align*} L_a u(x)&=2\operatorname{P.V.}\int_{\mathbb{R}^N}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{s,p}(x,y) &\ \ \ +2\operatorname{P.V.}\int_{\mathbb{R}^N}a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{t,q}(x,y)dy. \end{align*} We establish the equivalence between weak and viscosity solutions under boundedness and continuity assumptions. In addition, the local boundedness of weak solutions in some special cases on $f$ is also obtained using the notion of De Giorgi classes.