On the negative limit of viscosity solutions for discounted Hamilton-Jacobi equations

TL;DR

This paper studies the behavior of minimal viscosity solutions to discounted Hamilton-Jacobi equations on closed manifolds as the discount factor approaches zero, showing convergence to a unique solution and providing a dynamical interpretation.

Contribution

It proves the convergence of minimal viscosity solutions for generic Tonelli Hamiltonians as the discount parameter tends to zero and offers a new dynamical perspective on the limit solution.

Findings

Minimal viscosity solutions converge to a unique solution as discount vanishes

Convergence holds for generic Tonelli Hamiltonians on closed manifolds

Provides a dynamical interpretation of the limit solution

Abstract

Suppose $M$ is a closed Riemannian manifold. For a $C^2$ generic (in the sense of Ma\~n\'e) Tonelli Hamiltonian $H: T^*M\rightarrow\mathbb{R}$, the minimal viscosity solution $u_\lambda^-:M\rightarrow \mathbb{R}$ of the negative discounted equation \[-\lambda u+H(x,d_xu)=c(H),\quad x\in M,\ \lambda>0 \] with the Ma\~n\'e's critical value $c(H)$ converges to a uniquely established viscosity solution $u_0^-$ of the critical Hamilton-Jacobi equation \[ H(x,d_x u)=c(H),\quad x\in M \] as $\lambda\rightarrow 0_+$. We also propose a dynamical interpretation of $u_0^-$.