Calculus of Variations and Geometric Measure Theory

A. Davini - L. Wang

On the vanishing discount problem from the negative direction

created by davini on 12 Jul 2023


Published Paper

Inserted: 12 jul 2023
Last Updated: 12 jul 2023

Journal: Discrete Contin. Dyn. Syst.
Volume: 41
Number: 5
Pages: 2377–2389
Year: 2021

ArXiv: 2007.12458 PDF


It has been proved in 10 that the unique viscosity solution of \begin{equation}\label{abs}\tag{} \lambda u\lambda+H(x,dx 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.