Regularity of Hamilton-Jacobi equations when forward is backward

Abstract

We introduce a general principle determining in certain cases the regularity of the viscosity solutions of Hamilton-Jacobi equations. This principle says that if one can solve the equation forward in time from some initial data and then backward in time resulting in the same initial data, then the solution must be C1. Some cases are given when this holds as well as an example when it does not. Convexity of either the hamiltonian or the initial data plays a crucial role throughout.