Regularity for a class of degenerate fully nonlinear nonlocal elliptic equations

TL;DR

This paper studies the regularity of solutions to a broad class of degenerate fully nonlinear nonlocal elliptic equations, revealing how solution smoothness depends on the fractional diffusion order.

Contribution

It provides a comprehensive regularity characterization for viscosity solutions across different fractional diffusion orders, including new gradient regularity results near the critical order 2.

Findings

Hölder continuity of the gradient when diffusion order is close to 2

Existence of solutions with $C^{1, eta}$ regularity for diffusion order in (1,2)

Hölder continuity of solutions for diffusion order in (0,1]

Abstract

We consider a wide class of fully nonlinear integro-differential equations that degenerate when the gradient of the solution vanishes. By using compactness and perturbation arguments, we give a complete characterization of the regularity of viscosity solutions according to different diffusion orders. More precisely, when the order of the fractional diffusion is sufficiently close to 2, we obtain H\"{o}lder continuity for the gradient of any viscosity solutions and further derive an improved gradient regularity estimate at the origin. For the order of the fractional diffusion in the interval $(1, 2)$, we prove that there is at least one solution of class $C^{1, \alpha}_{\rm loc}$. Additionally, if the order of the fractional diffusion is in the interval $(0,1]$, the local H\"{o}lder continuity of solutions is inferred.