The Lax-Oleinik semigroup on graphs
Renato Iturriaga Héctor Sánchez Morgado
Networks & Heterogeneous Media 2017, 12(4): 643-662 doi: 10.3934/nhm.2017026

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
Physical solutions of the Hamilton-Jacobi equation
Nalini Anantharaman Renato Iturriaga Pablo Padilla Héctor Sánchez-Morgado
Discrete & Continuous Dynamical Systems - B 2005, 5(3): 513-528 doi: 10.3934/dcdsb.2005.5.513
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
Discrete & Continuous Dynamical Systems - B 2011, 15(3): 623-635 doi: 10.3934/dcdsb.2011.15.623
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.
Selection of calibrated subaction when temperature goes to zero in the discounted problem
Renato Iturriaga Artur O. Lopes Jairo K. Mengue
Discrete & Continuous Dynamical Systems - A 2018, 38(10): 4997-5010 doi: 10.3934/dcds.2018218

Consider $T(x) = d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, $zhongwenzy<\lambda<1$ and the unique function $b_\lambda:S^1 \to \mathbb{R}$ satisfying $ b_\lambda(x) = \max_{T(y) = x} \{ \lambda \, b_\lambda(y) + A(y)\}. $

We will show that, when $\lambda \to 1$, the function $b_\lambda- \frac{m(A)}{1-\lambda}$ converges uniformly to the calibrated subaction $V(x) = \max_{\mu \in \mathcal{ M}} \int S(y, x) \, d \mu(y)$, where $S$ is the Mañe potential, $\mathcal{ M}$ is the set of invariant probabilities with support on the Aubry set and $m(A) = \sup_{\mu \in \mathcal{M}} \int A\, d\mu$.

For $\beta>0$ and $\lambda \in (0, 1)$, there exists a unique fixed point $u_{\lambda, \beta} :S^1\to \mathbb{R}$ for the equation $e^{u_{\lambda, \beta}(x)} = \sum_{T(y) = x}e^{\beta A(y) +\lambda u_{\lambda, \beta}(y)}$. It is known that as $\lambda \to 1$ the family $e^{[u_{\lambda, \beta}- \sup u_{\lambda, \beta}]}$ converges uniformly to the main eigenfuntion $\phi_\beta $ for the Ruelle operator associated to $\beta A$. We consider $\lambda = \lambda(\beta)$, $\beta(1-\lambda(\beta))\to+\infty$ and $\lambda(\beta) \to 1$, as $\beta \to\infty$. Under these hypotheses we will show that $\frac{1}{\beta}(u_{\lambda, \beta}-\frac{P(\beta A)}{1-\lambda})$ converges uniformly to the above $V$, as $\beta\to \infty$. The parameter $\beta$ represents the inverse of temperature in Statistical Mechanics and $\beta \to \infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $\beta \to \infty$.

keywords: Calibrated subaction maximizing measure discounted method zero temperature limit eigenfunction of the Ruelle operator

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