A new selection problem for degenerate viscous Hamilton-Jacobi equations

TL;DR

This paper investigates a selection mechanism for solutions of degenerate viscous Hamilton-Jacobi equations, combining approximation methods and the nonlinear adjoint approach to identify a distinguished ergodic solution.

Contribution

It introduces a novel approximation scheme and a formula for selecting specific solutions of the ergodic problem using generalized Mather measures.

Findings

Established uniform convergence of approximations to a distinguished solution.

Derived an explicit formula for the selected limit involving Mather measures and potential.

Demonstrated the flexibility to realize any prescribed ergodic solution with convergence rate.

Abstract

We study a selection problem for degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians, in which the approximation procedure combines a nonlinear discounted approximation with a small potential perturbation. A key question is how their simultaneous effects influence the asymptotic selection of viscosity solutions of the associated ergodic problem. Based on the nonlinear adjoint method, we establish the uniform convergence of the approximating solutions to a distinguished solution of the ergodic problem and derive a formula for the selected limit in terms of generalized Mather measures and the potential. As an application, we show that this selection principle is sufficiently flexible to realize any prescribed solution of the ergodic problem, with an explicit convergence rate.