# Double obstacle problems and fully nonlinear PDE with non-strictly convex gradient constraints

## TL;DR

This paper establishes optimal second-order regularity for fully nonlinear elliptic equations with convex gradient constraints, even when constraints lack smoothness or strict convexity, by relating them to double obstacle problems.

## Contribution

It introduces a novel approach linking gradient-constrained equations to double obstacle problems and proves their optimal regularity without assuming regularity of the constraints.

## Key findings

- Optimal $W^{2,
abla}$ regularity for equations with convex gradient constraints.
- Regularity results extend up to the boundary.
- Connection between gradient constraints and double obstacle problems.

## Abstract

We prove the optimal $W^{2, \infty }$ regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal $W^{2, \infty }$ regularity for 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.