Regularity theory for degenerate fully nonlinear nonlocal equations with a Hamiltonian term

TL;DR

This paper studies a class of complex nonlocal equations with Hamiltonian terms, establishing regularity properties of solutions, including Lipschitz continuity, gradient Hölder continuity, and conditions for differentiability.

Contribution

It provides a detailed analysis of the regularity of solutions to degenerate fully nonlinear nonlocal equations with Hamiltonian terms, using novel characterization and perturbation techniques.

Findings

Proved Lipschitz regularity of viscosity solutions.

Established gradient Hölder continuity of solutions.

Explored conditions for $C^1$-differentiability under minimal assumptions.

Abstract

We investigate a class of degenerate fully nonlinear nonlocal elliptic equations with Hamiltonian terms. By precisely characterizing the interaction between the degeneracy law of equations and the growth behavior of the Hamiltonian terms, we establish the Lipschitz regularity of viscosity solutions by the Ishii-Lions method, and further show the gradient H\"{o}lder continuity for solutions via utilizing perturbation techniques. Additionally, under minimal assumptions on the degeneracy pattern, the $C^1$-differentiability property of solutions is explored as well.