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 C2 submanifold
of a Finsler manifold.