On the vanishing discount problem from the negative direction

Abstract

It has been proved in [10] that the unique viscosity solution of \begin{equation}\label{abs}\tag{*} λu_λ+H(x,d_x u_λ)=c(H)\qquad\hbox{in $M$}, \end{equation} uniformly converges, for $\lambda \to 0^{+}$, 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 $\lambda \to 0^{-}$. 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 $\lambda \to 0^{-}$. Furthermore, we exhibit an example of $H$ for which equation \eqref{abs} admits a unique solution for $\lambda <0$ as well.

14 pages

References in corpus (1)

• Convergence of the solutions of the discounted equation: the discrete case