ESAIM: Control, Optimisation and Calculus of Variations

Research Article

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

Mennucci, Andrea C.G.

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 $n-1$ dimensional manifolds, but for a ${{\mathcal H}}^{n-1}$ negligeable subset. These results can be applied to the cutlocus of a C 2 submanifold of a Finsler manifold.

(Received May 27 2003)

(Online publication June 15 2004)

Key Words:

  • Hamilton-Jacobi equations;
  • conjugate points.

Mathematics Subject Classification:

  • 49L25;
  • 53C22;
  • 53C60
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