Non-existence of global characteristics for viscosity solutions

TL;DR

This paper proves that for non-convex Hamiltonians, viscosity solutions do not share the same global characteristics as variational solutions, highlighting fundamental differences in solution behavior.

Contribution

It demonstrates that no broader class of Hamiltonians beyond convex ones can have viscosity solutions coinciding with variational solutions, using explicit counterexamples.

Findings

Viscosity solutions differ globally for non-convex Hamiltonians.

Constructed explicit examples showing the non-coincidence of solution graphs.

Analyzed saddle Hamiltonians and reduced cases to establish the main result.

Abstract

Two different types of generalized solutions, namely viscosity and variational solutions, were introduced to solve the first-order evolutionary Hamilton--Jacobi equation. They coincide if the Hamiltonian is convex in the momentum variable. In this paper we prove that there exists no other class of integrable Hamiltonians sharing this property. To do so, we build for any non-convex non-concave integrable Hamiltonian a smooth initial condition such that the graph of the viscosity solution is not contained in the wavefront associated with the Cauchy problem. The construction is based on a new example for a saddle Hamiltonian and a precise analysis of the one-dimensional case, coupled with reduction and approximation arguments.