Convergence of the solutions of the nonlinear discounted Hamilton-Jacobi equation: the central role of Mather measures

DOI10.1016/j.matpur.2023.10.006zbMath1530.35037arXiv2301.10478MaRDI QIDQ6187082

Qinbo Chen, Albert Fathi, Jian Lu Zhang, Maxime Zavidovique

Publication date: 10 January 2024

Published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2301.10478

zbMATH Keywords

comparison principleviscosity solutionsMather measuresdiscounted Hamilton-Jacobi equationsweak kam theory

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games (49L25) Viscosity solutions to PDEs (35D40) Hamilton-Jacobi equations (35F21) PDEs on manifolds (35R01)

Cites Work

• Convergence of the solutions of the discounted equation: the discrete case
• Limit of the infinite horizon discounted Hamilton-Jacobi equation
• Selection problems for a discount degenerate viscous Hamilton-Jacobi equation
• Convergence of the solutions of discounted Hamilton-Jacobi systems
• Action minimizing invariant measures for positive definite Lagrangian systems
• Perron's method for Hamilton-Jacobi equations
• Variational construction of connecting orbits
• Viscosity solutions of Hamilton-Jacobi equations
• Random Lax-Oleinik semigroups for Hamilton-Jacobi systems
• The selection problem for discounted Hamilton-Jacobi equations: some non-convex cases
• PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians
• Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation
• Weak KAM theory for nonregular commuting Hamiltonians
• On the vanishing discount problem from the negative direction
• Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations
• Convergence of solutions for some degenerate discounted Hamilton-Jacobi equations
• On the \(C^1\) and \(C^2\)-convergence to weak K.A.M. solutions
• The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. I: Linear coupling
• The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. II: Nonlinear coupling
• Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample
• The vanishing discount problem and viscosity Mather measures. 1: The problem on a torus
• The vanishing discount problem and viscosity Mather measures. II: Boundary value problems
• Aubry-Mather theory for conformally symplectic systems
• Long time behavior of the master equation in mean field game theory
• On the (non) existence of viscosity solutions of multi-time Hamilton-Jacobi equations
• Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function
• Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set
• The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space
• Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations
• Generalized Mather problem and selection principles for viscosity solutions and mather measures
• Viscosity Solutions of Hamilton-Jacobi Equations
• Generic properties and problems of minimizing measures of Lagrangian systems
• Vanishing contact structure problem and convergence of the viscosity solutions
• Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations
• On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case