Nonlocal fully nonlinear double obstacle problems

TL;DR

This paper establishes existence and regularity results for nonlocal fully nonlinear double obstacle problems, extending understanding of boundary behavior and solution smoothness without requiring smooth obstacles.

Contribution

It introduces a penalization method adapted for nonlocal equations to prove existence and regularity of solutions with less restrictive obstacle smoothness assumptions.

Findings

Proved existence of solutions with $C^{1,eta}$ regularity.

Established boundary regularity results.

Handled obstacles that are only Lipschitz semi-concave/semi-convex.

Abstract

We prove the existence and $C^{1,\alpha}$ regularity of solutions to nonlocal fully nonlinear elliptic double obstacle problems. We also obtain boundary regularity for these problems. The obstacles are assumed to be Lipschitz semi-concave/semi-convex functions, and we do not require them to be $C^1$. Our approach is to adapt a penalization method to be applicable to the setting of nonlocal equations and their viscosity solutions.