  We study the nonhomogeneous Dirichlet problem for first order Hamilton-Jacobi
equations associated with Tonelli Hamiltonians on a bounded domain $\Omega$ of
$\R^n$ assuming the energy level to be supercritical. First, we show that the
viscosity (weak KAM) solution of such a problem is Lipschitz continuous and
locally semiconcave in $\Omega$. Then, we analyse the singular set of a
solution showing that singularities propagate along suitable curves, the
so-called generalized characteristics, and that such curves stay singular
unless they reach the boundary of $\Omega$. Moreover, we prove that the latter
is never the case for mechanical systems and that singular generalized
characteristics converge to a critical point of the solution in finite or
infinite time. Finally, under stronger assumptions for the domain and Dirichlet
data, we are able to conclude that solutions are globally semiconcave and
semiconvex near the boundary.
