Nonlocal equations with gradient constraints

TL;DR

This paper establishes existence and regularity results for nonlocal fully nonlinear elliptic equations with gradient constraints, linking them to double obstacle problems and employing monotonicity properties.

Contribution

It introduces a novel approach connecting nonlocal equations with gradient constraints to double obstacle problems, proving regularity without assuming regularity of the constraints.

Findings

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

Boundary regularity of solutions is $C^{0,1}$.

Regularity results hold without smoothness assumptions on constraints.

Abstract

We prove the existence and $C^{1,\alpha}$ regularity of solutions to nonlocal fully nonlinear elliptic equations with gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or strictly convex. We also obtain $C^{0,1}$ boundary regularity for these problems. Our approach is to show that these nonlocal equations with gradient constraints are related to some nonlocal double obstacle problems. Then we prove the regularity of the double obstacle problems. In this process, we also employ the monotonicity property for the second derivative of obstacles, which we have obtained in a previous work.