Regularity of Solutions to Hamilton-Jacobi Equations

Abstract

We formulate a Hamilton-Jacobi partial differential equation H(x, Du(x)) = 0 on a n dimensional manifold M, with assumptions of uniform convexity of H(x,.)and regularity of H in a neighborhood of {H = 0} in T*M; we define the “min solution” u, a generalized solution, which often coincides with the viscosity solution; the definition is suited to proving regularity results about u; in particular, we prove that the closure of the set where u is not regular is a ℋ^n-1-rectifiable set.

Dedication: to Sanjoy, for the times he wasted listening to my wrong conjectures, for the times he spent listening to my right conjectures.

full version of this paper is a preprint of the Scuola Normale Superiore.