## Journals

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NHM

We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result is the long time convergence of the Lax Oleinik semi-group. It follows that weak KAM solutions are viscosity solutions of the Hamilton-Jacobi equation [

DCDS-B

We consider a Lagrangian system on the d-dimensional torus, and
the associated Hamilton-Jacobi equation. Assuming that the Aubry set of
the system consists in a finite number of hyperbolic periodic orbits of the
Euler-Lagrange flow, we study the vanishing-viscosity limit, from the viscous
equation to the inviscid problem. Under suitable assumptions, we show that
solutions of the viscous Hamilton-Jacobi equation converge to a unique solution
of the inviscid problem.

DCDS-B

Motivated by the infinite horizon discounted problem,
we study the convergence of solutions of the Hamilton Jacobi
equation when the discount vanishes. If the Aubry set consists in a
finite number of hyperbolic critical points, we give
an explicit expression for the limit. Additionaly, we give a new
characterization of Mañé's critical value as for wich the set
of viscosity solutions is equibounded.

keywords:
Hamilton-Jacobi equation.

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