On the vanishing discount problem from the negative direction

TL;DR

This paper investigates the behavior of minimal viscosity solutions of a Hamilton-Jacobi equation as the discount parameter approaches zero from the negative side, establishing convergence to the critical solution under certain conditions.

Contribution

It extends the vanishing discount problem analysis to negative discount parameters, proving convergence of minimal solutions to the critical solution.

Findings

Minimal solutions converge to the critical solution as rom the negative side.

Under certain assumptions, solutions are unique for or negative values.

Provides an example where the equation admits a unique solution for rom below.

Abstract

It has been proved in [10] that the unique viscosity solution of \begin{equation}\label{abs}\tag{*} \lambda u_\lambda+H(x,d_x u_\lambda)=c(H)\qquad\hbox{in $M$}, \end{equation} uniformly converges, for $\lambda\rightarrow 0^+$, to a specific solution $u_0$ of the critical equation \[ H(x,d_x u)=c(H)\qquad\hbox{in $M$}, \] where $M$ is a closed and connected Riemannian manifold and $c(H)$ is the critical value. In this note, we consider the same problem for $\lambda\rightarrow 0^-$. In this case, viscosity solutions of equation \eqref{abs} are not unique, in general, so we focus on the asymptotics of the minimal solution $u_\lambda^-$ of \eqref{abs}. Under the assumption that constant functions are subsolutions of the critical equation, we prove that the $u_\lambda^-$ also converges to $u_0$ as $\lambda\rightarrow 0^-$. Furthermore, we exhibit an example of $H$ for which equation \eqref{abs} admits a unique solution for $\lambda<0$ as well.