The vanishing discount problem and viscosity Mather measures. Part 1: the problem on a torus

Abstract

We develop a variational approach to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such partial differential equations, which are natural extensions of the Mather measures. Using the viscosity Mather measures, we prove that the whole family of solutions $v^{\lambda }$ of the discount problem with the factor $\lambda >0$ converges to a solution of the ergodic problem as $\lambda \to 0$.

23 pages, this is a revised version with corrections of minor errors and typos

References in corpus (5)

• Convergence of the solutions of the discounted equation
• Convergence of the solutions of the discounted equation: the discrete case
• The vanishing discount problem and viscosity Mather measures. Part 2: boundary value problems
• The Dirichlet problem for the Bellman equation at resonance
• Selection problems for a discounted degenerate viscous Hamilton--Jacobi equation

Cited by in corpus (16)

• Long time behavior of the master equation in mean-field game theory
• The vanishing discount problem and viscosity Mather measures. Part 2: boundary value problems
• Vanishing contact structure problem and convergence of the viscosity solutions
• Convergence of solutions for some degenerate discounted Hamilton--Jacobi equations
• Vanishing discount problem and the additive eigenvalues on changing domains
• Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function
• Limit of solutions for semilinear Hamilton-Jacobi equations with degenerate viscosity
• The Vanishing Discount problem for Hamilton-Jacobi Equations in the Euclidean Space
• Weak KAM theory for discount Hamilton-Jacobi equations and its application
• On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem
• On the regularity of stochastic effective Hamiltonian
• An example in the vanishing discount problem for monotone systems of Hamilton-Jacobi equations
• Exponential convergence of 1-graph of the solution semigroup of contact Hamilton-Jacobi equations
• Generalized ergodic problems: existence and uniqueness structures of solutions
• Twisted Lax--Oleinik formulas and weakly coupled systems of Hamilton--Jacobi equations
• Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains