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 Hölder regularity for bounded viscosity solutions under general structural and continuity assumptions on the underlying Lévy measures, without imposing any ellipticity condition on the nonlocal operator. The Hölder exponent is given explicitly in terms of the order σ ∈ (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 < σ < 2, and yields an improved regularity exponent when σ ∈ (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 Hölder 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 Lévy measures is merely Hölder continuous, thereby illustrating the sharpness of our continuity assumptions.
Related
- Potential estimates and improved Sobolev regularity for nonlocal fully nonlinear elliptic equations
- Global regularity for 3D large spherically symmetric degenerate compressible Navier-Stokes equations with superlinear BD viscosity and far-field vacuum
- Quantitative periodic homogenization and uniform regularity for nonlocal equations
- Hölder regularity for nonlocal equations governed by measures and non-standard growth
- Interior regularity of mixed local-nonlocal parabolic semilinear equations
BibTeX
@misc{ciomaga2026improved,
title = {{Improved Regularity for Nonlocal Hamilton--Jacobi Equations with Superlinear Hamiltonians}},
author = {Adina Ciomaga and Tri Minh Lê and Olivier Ley and Erwin Topp},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16918},
eprint = {2609.16918},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05737933v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Improved Regularity for Nonlocal Hamilton--Jacobi Equations with Superlinear Hamiltonians
AU - Adina Ciomaga
AU - Tri Minh Lê
AU - Olivier Ley
AU - Erwin Topp
PY - 2026
DA - 2026/09/03
PB - HAL
DO - 10.48550/arxiv.2609.16918
UR - https://hal.science/hal-05737933v1
AB - 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 Hölder regularity for bounded viscosity solutions under general structural and continuity assumptions on the underlying Lévy measures, without imposing any ellipticity condition on the nonlocal operator. The Hölder exponent is given explicitly in terms of the order σ ∈ (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 < σ < 2, and yields an improved regularity exponent when σ ∈ (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 Hölder 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 Lévy measures is merely Hölder continuous, thereby illustrating the sharpness of our continuity assumptions.
ER -