论文标题
消失的折扣问题和变化域上的添加剂特征值
Vanishing discount problem and the additive eigenvalues on changing domains
论文作者
论文摘要
我们研究了状态强的汉密尔顿 - 雅各比方程$ ϕ（λ）u_λ（x）+h（x，x，du_λ（x））= 0 $ in $（1+r（λ））ω$和相应的添加剂eigenvalues $ c（x）（x x（x），在$（1+r（λ））中，带有状态约束。在这里，$ω$是$ \ mathbb {r}^n $，$ ϕ（λ），r（λ）:( 0，\ infty）\ rightArrow \ rightarrow \ mathbb {r} $是连续的函数，因此$ demangative and nonegative and $ \ lim_ {λ\ right a = 0^0^0^0^+} \ lim_ {λ\ rightarrow 0^+} r（λ）= 0 $。我们在凸环设置中获得收敛和非连接结果。此外，我们为添加剂特征值$ c（λ）$的渐近扩展为$λ\ rightarrow 0^+$提供了最初的结果。我们使用的主要工具是用粘度马瑟测量的溶液的双重表示。
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.
