Global optimal regularity for variational problems with nonsmooth non-strictly convex gradient constraints

We prove the optimal W(2,infinity )regularity for variational problems with convex gradient constraints. We do not assume any regularity of the constraints; so the constraints can be nonsmooth, and they need not be strictly convex. When the domain is smooth enough, we show that the optimal regularity holds up to the boundary. In this process, we also characterize the set of singular points of the viscosity solutions to some Hamilton-Jacobi equations. Furthermore, we obtain an explicit formula for the second derivative of these viscosity solutions; and we show that the second derivatives satisfy a monotonicity property. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This study demonstrates the optimal W(2,infinity) regularity for variational problems with convex gradient constraints and characterizes the set of singular points of the viscosity solutions to some Hamilton-Jacobi equations. Explicit formula for the second derivative of the viscosity solutions is obtained, showing that the second derivatives exhibit a monotonicity property.