COMPARISON PRINCIPLE FOR GENERAL NONLOCAL HAMILTON-JACOBI EQUATIONS WITH SUPERLINEAR GRADIENT
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 Lévy 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, Lévy-Itô operators).
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