Comparison principle for general nonlocal Hamilton-Jacobi equations with superlinear gradient

TL;DR

This paper establishes a comparison principle for discontinuous viscosity solutions of nonlocal Hamilton-Jacobi equations with superlinear gradient terms, broadening the scope of PDEs where uniqueness and stability can be analyzed.

Contribution

It introduces a comparison principle for a wide class of nonlocal Hamilton-Jacobi equations with minimal measure restrictions, including degenerate and variable order operators.

Findings

Comparison principle proven for discontinuous viscosity solutions

Applicable to various nonlocal operators including Levy and variable order

Weakest measure conditions used in viscosity approach for these PDEs

Abstract

We obtain the comparison principle for discontinuous viscosity sub- and supersolutions of nonlocal Hamilton-Jacobi equations, with superlinear and coercive gradient terms. The nonlocal terms are integro-differential operators in L\'evy form, with general measures: $x$-dependent, possibly degenerate and without any restriction on the order. The measures must satisfy a combined Wasserstein/Total Variation-continuity assumption, which is one of the weakest conditions used in the context of viscosity approach for this type of integro-differential PDEs. The proof relies on a regularizing effect due to the gradient growth. We present several examples of applications to PDEs with different types of nonlocal operators (measures with density, operators of variable order, L\'evy-It\^o operators).