We formulate an Hamilton-Jacobi partial differential equation$$H(x,Du(x\left)\right)=0$$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 “min solution” $u$, a generalized solution; to this end, we view $T^{*}M$ as a symplectic manifold. The definition of “min 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...

This errata corrects one error in the 2004 version of this paper [Mennucci, ESAIM: COCV10 (2004) 426–451].

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...

We discuss differentiability properties of convex functions on Heisenberg groups. We show that the notions of horizontal convexity (h-convexity) and viscosity convexity (v-convexity) are equivalent and that h-convex functions are locally Lipschitz continuous. Finally we exhibit Weierstrass-type h-convex functions which are nowhere differentiable in the vertical direction on a dense set or on a Cantor set of vertical lines.

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40],...