Improved Regularity for Nonlocal Hamilton--Jacobi Equations with Superlinear Hamiltonians

Abstract

We investigate the regularity of viscosity solutions to a class of nonlocal Hamilton-Jacobi equations driven by x-dependent integro-differential operators and coercive superlinear Hamiltonians. We first establish H{\"o}lder regularity for bounded viscosity solutions under general structural and continuity assumptions on the underlying L{\'e}vy measures, without imposing any ellipticity condition on the nonlocal operator. The H{\"o}lder exponent is given explicitly in terms of the order $\sigma$ $\in$ (0, 2) of the operator and the growth exponent m>1 of the Hamiltonian. In particular, our approach applies to arbitrary superlinear Hamiltonians, including the delicate regime 1<m<$\sigma$<2, and yields an improved regularity exponent when $\sigma$ $\in$ (1, 2). Assuming in addition a weak ellipticity condition on the nonlocal operator, we prove that viscosity solutions are globally Lipschitz continuous. The proof combines the H{\"o}lder regularity supplied by the coercive Hamiltonian with the regularizing effect of the nonlocal diffusion through an Ishii-Lions argument, allowing us to treat Hamiltonians with arbitrary superlinear growth. Finally, we provide a counterexample showing that, in the absence of ellipticity, Lipschitz regularity may fail if the spatial dependence of the L{\'e}vy measures is merely H{\"o}lder continuous, thereby illustrating the sharpness of our continuity assumptions.