Vanishing discount problem and the additive eigenvalues on changing domains

Abstract

We study the asymptotic behavior, as $λ\rightarrow 0^+$, of the state-constraint Hamilton--Jacobi equation $ϕ(λ) u_λ(x) + H(x,Du_λ(x)) = 0$ in $(1+r(λ))Ω$ and the corresponding additive eigenvalues, or ergodic constant $H(x,Dv(x)) = c(λ)$ in $(1+r(λ))Ω$ with state-constraint. Here, $Ω$ is a bounded domain of $ \mathbb{R}^n$, $ϕ(λ), r(λ):(0,\infty)\rightarrow \mathbb{R}$ are continuous functions such that $ϕ$ is nonnegative and $\lim_{λ\rightarrow 0^+} ϕ(λ) = \lim_{λ\rightarrow 0^+} r(λ) = 0$. We obtain both convergence and non-convergence results in the convex setting. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue $c(λ)$ as $λ\rightarrow 0^+$. The main tool we use is a duality representation of solution with viscosity Mather measures.