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

TL;DR

This paper establishes the optimal second-order regularity for variational problems with convex, possibly nonsmooth, gradient constraints, extending regularity results to nonsmooth and non-strictly convex constraints and characterizing singularities.

Contribution

It proves the $W^{2, abla}$ regularity without assuming smoothness or strict convexity of the constraints, and characterizes singular points of viscosity solutions.

Findings

Optimal $W^{2, abla}$ regularity for nonsmooth constraints

Explicit second derivative formula for viscosity solutions

Monotonicity property of second derivatives

Abstract

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