PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians
type: Item
instance of: scholarly article
title: PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians
zbMATH Open document ID: 1065.35092
author: Albert Fathi
author: Antonio Siconolfi
published in: Calculus of Variations and Partial Differential Equations
publication date: 2005-02-21T00:00:00Z
review text: The main purpose of this paper is to generalize the Aubry-Mather theory for Hamiltonians \(H\), defined on the cotangent bundle of the flat torus \(\mathbb T^N\), satisfying weak regularity assumptions. More precisely, in the first part of the paper \(H\) is coercive quasiconvex and continuous, and in the second part a strict convexity as well as a local Lipschitz-continuity assumption is added. In such a setting, no Hamiltonian dynamics can necessarily be defined, at least in the usual sense, and consequently dynamical systems techniques are not suitable. The authors attack the problem using PDE methods, and in particular the theory of viscosity solutions. Then, the family of Hamilton-Jacobi equations \(H(x,Du)= a\) on \(\mathbb T^N\), with \(a\) a real parameter, is considered. It is well known that a special value of \(a\) is relevant, qualified as critical and denoted in the remainder by \(c\). It is characterized, from a PDE viewpoint, by the property that the corresponding equation can be solved in the viscosity sense on the whole torus. Finally, a generalization of the projected Mather set, under the restriction that the set of equilibria is empty, is provided and it is proved that it is a uniqueness set for the equation \(H(x, Du)= c\).
reviewed by: Vasile Iftode
Mathematics Subject Classification ID: 35F20
Mathematics Subject Classification ID: 37J50
Mathematics Subject Classification ID: 49K10
zbMATH DE Number: 2135760
zbMATH Keywords: Hamilton-Jacobi equations
zbMATH Keywords: viscosity solutions
MaRDI profile type: Publication
full work available at URL: s00526-004-0271-z
OpenAlex ID: W2086763898
cites work: Q3324260
cites work: Viscosity solutions of Hamilton-Jacobi equations
cites work: Q4210481
cites work: Optimization and nonsmooth analysis
cites work: Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems
cites work: Hamilton-Jacobi equations with measurable dependence on the state variable.
cites work: Semiconcave functions, Hamilton-Jacobi equations, and optimal control
cites work: Q4944845
cites work: Periodic homogenisation of certain fully nonlinear partial differential equations
cites work: Solutions KAM faibles conjuguées et barrières de Peierls
cites work: Sur la convergence du semi-groupe de Lax-Oleinik
cites work: Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens
cites work: Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation
cites work: Q4229034
cites work: Q3202346
cites work: Q3961832
cites work: Metric character of Hamilton–Jacobi equations
cites work: Semiconcavity of solutions of stationary Hamilton-Jacobi equations
DOI: 10.1007/S00526-004-0271-Z
description: scientific article; zbMATH DE number 2135760