Regularity estimates for fully nonlinear integro-differential equations with nonhomogeneous degeneracy

TL;DR

This paper studies the regularity of solutions to a class of fully nonlinear nonlocal equations with degeneracy or singularity, establishing existence and regularity results in various function spaces.

Contribution

It provides new regularity estimates for solutions of degenerate and singular fully nonlinear integro-differential equations, including existence of $C_{loc}^{1, eta}$ solutions.

Findings

Existence of $C_{loc}^{1, eta}$ solutions in degenerate cases

Hölder regularity estimates for solutions under certain conditions

Gradient Hölder regularity in the singular setting

Abstract

We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class $C_{loc}^{1, \alpha}$, for some constant $\alpha \in (0, 1)$. In addition, under suitable conditions on $\sigma$, we prove regularity estimates in H\"older spaces for any viscosity solution. We also examine the singular setting and prove H\"older regularity estimates for the gradient of the solutions.