*Published Paper*

**Inserted:** 12 jul 2023

**Last Updated:** 12 jul 2023

**Journal:** Invent. Math.

**Volume:** 206

**Number:** 1

**Pages:** 29–55

**Year:** 2016

**Abstract:**

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 \[ \lambda u_{\lambda}(x)+H(x,d_x u_{\lambda})=c(H)\qquad\hbox{in $M$}, \] 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 \[ H(x,d_xu)=c(H)\qquad\hbox{in $M$}. \] We characterize $u_0$ in terms of Peierls barrier and projected Mather measures.