On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case

TL;DR

This paper investigates the asymptotic behavior of viscosity solutions to a perturbed Hamilton-Jacobi equation in one dimension as the discount factor approaches zero, establishing convergence to a specific solution linked to Mather measures and the unperturbed problem.

Contribution

It extends the vanishing discount approximation results to noncompact settings with compactly supported perturbations, connecting solutions to Mather measures and weak KAM theory.

Findings

Viscosity solutions converge locally uniformly as discount tends to zero.

The limit solution is characterized via projected Mather measures.

Results extend previous work to noncompact perturbations.

Abstract

We study the asymptotic behavior of the viscosity solutions $u^\lambda_G$ of the Hamilton-Jacobi (HJ) equation \begin{equation*} \lambda u(x)+G(x,u')=c(G)\qquad\hbox{in $\mathbb{R}$} \end{equation*} as the positive discount factor $\lambda$ tends to 0, where $G(x,p):=H(x,p)-V(x)$ is the perturbation of a Hamiltonian $H\in C({\mathbb R}\times{\mathbb R})$, ${\mathbb Z}$-periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential $V\in {C}_c({\mathbb R})$. The constant $c(G)$ appearing above is defined as the infimum of values $a\in {\mathbb R}$ for which the HJ equation $G(x,u')=a$ in ${\mathbb R}$ admits bounded viscosity subsolutions. We prove that the functions $u^\lambda_G$ locally uniformly converge, for $\lambda\rightarrow 0^+$, to a specific solution $u_G^0$ of the critical equation \begin{equation}\label{abs}\tag{*} G(x,u')=c(G)\qquad\hbox{in ${\mathbb R}$}. \end{equation} We identify $u^0_G$ in terms of projected Mather measures for $G$ and of the limit $u^0_H$ to the unperturbed periodic problem. This can be regarded as an extension to a noncompact setting of the main results in [17]. Our work also includes a qualitative analysis of \eqref{abs} with a weak KAM theoretic flavor.