Global subanalytic solutions of Hamilton–Jacobi type equations

Abstract

In the 80's, Crandall and Lions introduced the concept of viscosity solution, in order to get existence and/or uniqueness results for Hamilton–Jacobi equations. In this work, we first investigate the Dirichlet and Cauchy–Dirichlet problems for such equations, where the Hamiltonian is associated to a problem of calculus of variations, and prove that, if the data are analytic, then the viscosity solution is moreover subanalytic. We then extend this result to Hamilton–Jacobi equations stemming from optimal control problems, in particular from sub-Riemannian geometry, which are generalized eikonal equations.

As a consequence, the set of singularities of the viscosity solutions of such Hamilton–Jacobi equations is a subanalytic stratified manifold of codimension greater than or equal to one.