SBV regularity for Hamilton-Jacobi equations in $\mathbb R^n$

TL;DR

This paper proves that for uniformly convex Hamiltonians, the spatial gradient and time derivative of viscosity solutions to Hamilton-Jacobi equations are locally special functions of bounded variation, indicating a certain regularity.

Contribution

It establishes SBV regularity for derivatives of viscosity solutions under uniform convexity assumptions on the Hamiltonian.

Findings

$D_x u$ and $rac{ ext{d}u}{ ext{d}t}$ are in SBV_{loc}( ext{Omega})

Regularity result applies to solutions with uniformly convex Hamiltonians

Advances understanding of solution structure in Hamilton-Jacobi equations

Abstract

In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$ \partial_t u + H(D_{x} u)=0 \qquad \textrm{in} \Omega\subset \mathbb R\times \mathbb R^{n} . $$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb R^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(\Omega)$.