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

Stefano Bianchini, Camillo De Lellis, Roger Robyr

Published 2010-02-22, updated 2014-12-16Version 2

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 $SBV_{loc}(\Omega)$.