Physical solutions of the Hamilton-Jacobi equation
Nalini Anantharaman Renato Iturriaga Pablo Padilla Héctor Sánchez-Morgado
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.
keywords: Hamilton-Jacobi equation viscosity solution Aubry-Mather set.
Limit of the infinite horizon discounted Hamilton-Jacobi equation
Renato Iturriaga Héctor Sánchez-Morgado
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.
The Lax-Oleinik semigroup on graphs
Renato Iturriaga Héctor Sánchez Morgado

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 [3, 4], and in the case of Hamiltonians called of eikonal type in [3], we prove that the converse holds.

keywords: Lax-Oleinik semigroup weak KAM solution viscosity solution

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