  We consider a continuous coercive Hamiltonian $H$ on the cotangent bundle of
the compact connected manifold $M$ which is convex in the momentum. If
$u_\lambda:M\to\mathbb R$ is the viscosity solution of the discounted equation
$$
  \lambda u_\lambda(x)+H(x,d_x u_\lambda)=c(H), $$ where $c(H)$ is the critical
value, we prove that $u_\lambda$ converges uniformly, as $\lambda\to 0$, to a
specific solution $u_0:M\to\mathbb R$ of the critical equation $$ H(x,d_x
u)=c(H). $$ We characterize $u_0$ in terms of Peierls barrier and projected
Mather measures.

Analysis of PDEs