scholarly journals Weak KAM Solutions of a Discrete-Time Hamilton-Jacobi Equation in a Minimax Framework

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Porfirio Toledo
Keyword(s):  
Discrete Time ◽  
Jacobi Equation ◽  
Kam Theory ◽  
Infinite Time ◽  
Discount Factors ◽  
Long Time ◽  
Lax Operator ◽  
Discrete Time Case ◽  
Zero Sum ◽  
Hamilton Jacobi Equation

The purpose of this paper is to study the existence of solutions of a Hamilton-Jacobi equation in a minimax discrete-time case and to show different characterizations for a real number called the critical value, which plays a central role in this work. We study the behavior of solutions of this problem using tools of game theory to obtain a “fixed point” of the Lax operator associated, considering some facts of weak KAM theory to interpret these solutions as discrete viscosity solutions. These solutions represent the optimal payoff of a zero-sum game of two players, with increasingly long time payoffs. The developed techniques allow us to study the behavior of an infinite time game without using discount factors or average actions.

scholarly journals On weak KAM theory for N-body problems

2011 ◽  
Vol 32 (3) ◽  
pp. 1019-1041 ◽  
Author(s):  
EZEQUIEL MADERNA
Keyword(s):  
Configuration Space ◽  
Jacobi Equation ◽  
Kam Theory ◽  
Kam Theorem ◽  
Invariant Solutions ◽  
Weak Kam Theory ◽  
Minimal Action ◽  
Newtonian Case ◽  
Critical Action ◽  
Hamilton Jacobi Equation

AbstractWe consider N-body problems with potential 1/r2κ, where κ∈(0,1), including the Newtonian case (κ=1/2). Given R>0 and T>0, we find a uniform upper bound for the minimal action of paths binding, in time T, any two configurations which are contained in some ball of radius R. Using cluster partitions, we obtain from these estimates the Hölder regularity of the critical action potential (i.e. of the minimal action of paths binding two configurations in free time). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax–Oleinik semigroup, and we show that they are global viscosity solutions of the corresponding Hamilton–Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.

scholarly journals The Lax–Oleinik semi-group: a Hamiltonian point of view

2012 ◽  
Vol 142 (6) ◽  
pp. 1131-1177 ◽  
Author(s):  
Patrick Bernard
Keyword(s):  
Hamiltonian Systems ◽  
Viscosity Solutions ◽  
Jacobi Equation ◽  
Kam Theory ◽  
Companion Paper ◽  
Point Of View ◽  
Invariant Sets ◽  
Evolution Operators ◽  
Weak Kam Theory ◽  
Hamilton Jacobi Equation

The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on ℝd and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications.

scholarly journals On a model of a population with variable motility

2015 ◽  
Vol 25 (10) ◽  
pp. 1961-2014 ◽  
Author(s):  
Olga Turanova
Keyword(s):  
Diffusion Equation ◽  
Long Range ◽  
Viscosity Solutions ◽  
Jacobi Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Reaction Term ◽  
Long Time ◽  
Nonlocal Reaction ◽  
Hamilton Jacobi Equation

We study a reaction–diffusion equation with a nonlocal reaction term that models a population with variable motility. We establish a global supremum bound for solutions of the equation. We investigate the asymptotic (long-time and long-range) behavior of the population. We perform a certain rescaling and prove that solutions of the rescaled problem converge locally uniformly to zero in a certain region and stay positive (in some sense) in another region. These regions are determined by two viscosity solutions of a related Hamilton–Jacobi equation.

Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set

2012 ◽  
Vol 142 (6) ◽  
pp. 1193-1236 ◽  
Author(s):  
Albert Fathi
Keyword(s):  
Prior Knowledge ◽  
Jacobi Equation ◽  
Kam Theory ◽  
General Setting ◽  
Point Of View ◽  
Weak Kam Theory ◽  
First Order ◽  
Aubry Set ◽  
The Subject ◽  
Hamilton Jacobi Equation

We introduce the notion of a viscosity solution for the first-order Hamilton–Jacobi equation, in the more general setting of manifolds, to obtain a weak KAM theory using only tools from partial differential equations. This work should be accessible to people with no prior knowledge of the subject.

The Hamilton-Jacobi Equation and Weak KAM Theory

2017 ◽  
Author(s):  
Alfonso Sorrentino
Keyword(s):  
Jacobi Equation ◽  
Kam Theory ◽  
Invariant Sets ◽  
Classical Solutions ◽  
Kam Tori ◽  
Weak Kam Theory ◽  
Mather Theory ◽  
Starting Point ◽  
Interesting Approach ◽  
Hamilton Jacobi Equation

This chapter describes another interesting approach to the study of invariant sets provided by the so-called weak KAM theory, developed by Albert Fathi. This approach can be considered as the functional analytic counterpart of the variational methods discussed in the previous chapters. The starting point is the relation between KAM tori (or more generally, invariant Lagrangian graphs) and classical solutions and subsolutions of the Hamilton–Jacobi equation. It introduces the notion of weak (non-classical) solutions of the Hamilton–Jacobi equation and a special class of subsolutions (critical subsolutions). In particular, it highlights their relation to Aubry–Mather theory.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

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