arXiv subjects
Publications and source records attributed to Renato Iturriaga.
We extend some results of weak KAM theory to Lagrangians that are defined only on the horizontal distribution of a subriemannian manifold and depend on the unknown function.
We consider several $N$-body problems. The main result is a very simple and natural criterion for decoupling the Jacobi equation for some classes of them. If $E$ is a Euclidean space, and the potential function $U(x)$ for the $N$-body problem is a $C^2$ function defined in an open subset of $E^N$, then the Jacobi equation along a given motion $x(t)$ writes $\ddot J=HU_x(J)$, where the endomorphism $HU_x$ of $E^N$ represents the second derivative of the potential with respect to the mass inner product. Our splitting in particular applies to the case of homographic motions by central configurations. It allows then to deduce the well known Meyer-Schmidt decomposition for the linearization of the Euler-Lagrange flow in the phase space, formulated twenty years ago to study the relative equilibria of the planar $N$-body problem. However, our decomposition principle applies in many other classes of $N$-body problems, for instance to the case of isosceles three body problem, in which Sitnikov proved the existence of oscillatory motions. As a first concrete application, for the classical three-body problem we give a simple and short proof of a theorem of Y. Ou, ensuring that if the masses verify $μ=(m_1+m_2+m_3)^2/(m_1m_2+m_2m_3+m_1m_3)<27/8$ then the elliptic Lagrange solutions are linearly unstable for any value of the excentricity.
This article focuses two issues related to the first-order discounted mean field games system. The first is the time discretization problem. The time discretization approach enables us to prove the existence of solutions (u,m) of the system, where u is a viscosity solution of the discounted Hamilton-Jacobi equation and m is a projected minimizing measure satisfying the continuity equation in the sense of distributions. The second is the vanishing discount problems for both the discounted mean field games system and its discretized system. The methods we use primarily derive from weak KAM theory. Moreover, we provide an example demonstrating the non-uniqueness of solutions to the discounted mean field games system.
We provide an approximation scheme for first-order stationary mean field games with a separable Hamiltonian. First, we discretize Hamilton-Jacobi equations by discretizing in time, and then prove the existence of minimizing holonomic measures for mean field games. At last, we obtain two sequences of solutions $\{u_i\}$ of discrete Hamilton-Jacobi equations and minimizing holonomic measures $\{m_i\}$ for mean field games and show that $(u_i,m_i)$ converges to a solution of the stationary mean field games.
We consider a stochastic discretization of the stationary viscous Hamilton Jacobi equation on the flat d dimensional torus, associated with a Hamiltonian, convex and superlinear in the momentum variable. We show that each discrete problem admits a unique continuous solution on the torus, up to additive constants. By additionally assuming a technical condition on the associated Lagrangian, we show that each solution of the viscous Hamilton Jacobi equation is the limit of solutions of the discrete problems, as the discretization step goes to zero.
Consider $T(x)= d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, $0<λ<1$ and the unique function $b_λ:S^1 \to \mathbb{R}$ satisfying $ b_λ(x) = \max_{T(y)=x} \{ λ\, b_λ(y) + A(y)\}.$ We will show that, when $λ\to 1$, the function $b_λ- \frac{m(A)}{1-λ}$ converges uniformly to the calibrated subaction $V(x) = \max_{μ\in \mathcal{ M}} \int S(y,x) \, d μ(y)$, where $S$ is the Mañe potential, $\mathcal{ M}$ is the set of invariant probabilities with support on the Aubry set and $m(A)= \sup_{μ\in \mathcal{M}} \int A\,dμ$. For $β>0$ and $λ\in (0,1)$, there exists a unique fixed point $u_{λ,β} :S^1\to \mathbb{R}$ for the equation $e^{u_{λ,β}(x)} = \sum_{T(y)=x}e^{βA(y) +λu_{λ,β}(y)}$. It is known that as $λ\to 1$ the family $e^{[u_{λ,β}- \sup u_{λ,β}]}$ converges uniformly to the main eigenfuntion $ϕ_β$ for the Ruelle operator associated to $βA$. We consider $λ=λ(β)$, $β(1-λ(β))\to+\infty$ and $λ(β) \to 1$, as $β\to\infty$. Under these hypothesis we will show that $\frac{1}β(u_{λ,β}-\frac{P(βA)}{1-λ})$ converges uniformly to the above $V$, as $β\to \infty$. The parameter $β$ represents the inverse of temperature in Statistical Mechanics and $β\to \infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $β\to \infty$.
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. Weak KAM solutions are viscosity solutions, and in the case of Hamiltonians called of eikonal type in [CS], we prove that the converse holds.
We show that for a family of randomly kicked Hamiton-Jacobi equations on the torus, almost surely, the solution of an initial value problem converges exponentially fast to the unique stationary solution. Combined with the results in \cite{IK03} and \cite{KZ12}, this completes the program started in \cite{EKMS00} for the multi-dimensional setting.
We prove that, for generic (open and dense) values of the masses, the Newtonian potential function of the collinear N-body problem has $N!/2$ critical values when restricted to a fixed inertia level. In particular, we prove that for generic masses, there is only one minimal Moulton configuration.
We consider a continuous coercive Hamiltonian $H$ on the cotangent bundle of the compact connected manifold $M$ which is convex in the momentum. If $u_λ:M\to\mathbb R$ is the viscosity solution of the discounted equation $$ λu_λ(x)+H(x,d_x u_λ)=c(H), $$ where $c(H)$ is the critical value, we prove that $u_λ$ converges uniformly, as $λ\to 0$, 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.
We describe a setting for homogenization of convex hamiltonians on abelian covers of any compact manifold. In this context we also provide a simple variational proof of standard homogenization results.
In this work we prove the existence of Fathi's weak KAM solutions for periodic Lagrangians and give a construction of all of them.