Viscosity solutions for nonlocal equations with space-dependent operators

TL;DR

This paper studies viscosity solutions for elliptic and parabolic nonlocal equations with space-dependent fractional operators, establishing existence, uniqueness, spectral properties, and long-term behavior of solutions.

Contribution

It introduces a framework for viscosity solutions of nonlocal equations with variable domain operators and analyzes their spectral and asymptotic properties.

Findings

Unique solvability of elliptic and parabolic problems in viscosity sense

Existence and simplicity of the first eigenvalue of the elliptic operator

Parabolic solutions converge to elliptic solutions over time

Abstract

We consider a class of elliptic and parabolic problems, featuring a specific nonlocal operator of fractional-laplacian type, where integration is taken on variable domains. Both elliptic and parabolic problems are proved to be uniquely solvable in the viscosity sense. Moreover, some spectral properties of the elliptic operator are investigated, proving existence and simplicity of the first eigenvalue. Eventually, parabolic solutions are proven to converge to the corresponding limiting elliptic solution in the long-time limit.