Convergence of solutions for some degenerate discounted Hamilton--Jacobi equations

TL;DR

This paper investigates the convergence of solutions to a class of degenerate discounted Hamilton--Jacobi equations as the discount parameter approaches zero, establishing conditions under which solutions converge to a critical solution.

Contribution

The paper proves convergence of solutions for degenerate discounted Hamilton--Jacobi equations under specific conditions on the degeneracy function.

Findings

Solutions $u_$ converge to a critical solution $u_0$ as $ o 0$.

Convergence holds under suitable conditions on the degeneracy function $(x)$.

Provides a rigorous framework for understanding degenerate Hamilton--Jacobi equations.

Abstract

We study solutions of Hamilton--Jacobi equations of the form $$\lambda \alpha(x) u_\lambda(x) + H(x, D_x u_\lambda) = c,$$ where $\alpha$ is a nonnegative function, $\lambda$ a positive constant, $c$ a constant and $H $ a convex coercive Hamiltonian. Under suitable conditions on $\alpha$ we prove that the functions $u_\lambda$ converge as $\lambda\to 0$ to a function $u_0$ that is a solution of the critical equation $H(x, D_x u_0) = c$.