Nonlocal equations with gradient 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.

31 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1907.04683, arXiv:2105.09417, arXiv:1807.01590

References in corpus (2)

• Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited
• Nonlocal fully nonlinear double obstacle problems