## ESAIM: Control, Optimisation and Calculus of Variations

### Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity

Scuola Normale Superiore Piazza dei Cavalieri 7, 56126 Pisa, Italy; a.mennuci@sns.it.

Abstract

We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*M as a symplectic manifold. The definition of “minsol solution” is suited to proving regularity results about u; in particular, we prove in the first part that the closure of the set where u is not regular may be covered by a countable number of dimensional manifolds, but for a negligeable subset. These results can be applied to the cutlocus of a C 2 submanifold of a Finsler manifold.