Convergence of solutions for some degenerate discounted 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 $H(x,D_x{u}_0)=c$.

Version 2 ; 25 pages, modified according to the comments and suggestions by referees to improve presentation. Accepted for publication in Analysis & PDE

References in corpus (4)

• Weak KAM theorem on non compact manifolds
• Convergence of the solutions of the discounted equation: the discrete case
• Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample
• Convergence of viscosity solutions of generalized contact Hamilton-Jacobi equations

Cited by in corpus (3)

• Limit of solutions for semilinear Hamilton-Jacobi equations with degenerate viscosity
• Parameterized viscosity solutions of convex Hamiltonian systems with time periodic damping
• Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains