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

We prove the optimal $W^{2,\mathrm{\infty }}$ 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 $W^{2,\mathrm{\infty }}$ 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.