Double Obstacle Problems with obstacles given by non-$C^2$ Hamilton-Jacobi equations

TL;DR

This paper establishes optimal regularity results for double obstacle problems where obstacles are solutions to non-$C^2$ Hamilton-Jacobi equations, introducing a new pointwise regularity theory using non-homogeneous scaling.

Contribution

It develops a novel pointwise regularity theory for Hamilton-Jacobi equations with non-$C^2$ obstacles, overcoming limitations of standard techniques and proving optimal regularity.

Findings

Solutions are $C^{1,rac{eta}{2}}$ when $a \

in $C^eta$

Standard Bernstein techniques fail for non-$C^2$ obstacles, requiring new methods

Abstract

We prove optimal regularity for the double obstacle problem when obstacles are given by solutions to Hamilton-Jacobi equations that are not $C^2$. When the Hamilton-Jacobi equation is not $C^2$ then the standard Bernstein technique fails and we loose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speed in different directions) we develop a new pointwise regularity theory for Hamilton-Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that $C^1$-solutions to the Hamilton-Jacobi equation $$ \pm |\nabla h-a(x)|^2=\pm 1 \textrm{in} B_1, \qquad h=f \textrm{on} \partial B_1, $$ are in fact $C^{1,\alpha/2}$ provided that $a \in C^\alpha$. This result is optimal and to the authors' best knowledge new.