论文标题
关于从负面方向消失的折扣问题
On the vanishing discount problem from the negative direction
论文作者
论文摘要
在[10]中已经证明了\ begin {equion} \ label {abs} \ tag {*}λu_λ+h（x，x，d_xu_λ）= c（h）\ qquad \ qquad \ hbox {in $ m $}，$ niperly commiently commient，$ cemporter a $ cemporter，关键方程的$ u_0 $ \ [h（x，d_x u）= c（h）\ qquad \ hbox {in $ m $}，\]，其中$ m $是封闭且连接的riemannian歧管，$ c（h）$是关键值。在本说明中，我们考虑$λ\ rightarrow 0^ - $的同一问题。在这种情况下，方程\ eqref {abs}的粘度解决方案通常不是唯一的，因此，我们专注于最小解决方案的渐近学$u_λ^ - \ eqref {abs}的$u_λ^ - $。在假设常数函数是关键方程的亚物种的假设下，我们证明$u_λ^ - $也收敛到$ u_0 $，为$λ\ rightarrow 0^ - $。此外，我们展示了$ h $的示例，该示例的方程\ eqref {abs}也接受了$λ<0 $的唯一解决方案。
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 $λ\rightarrow 0^+$, 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 $λ\rightarrow 0^-$. In this case, viscosity solutions of equation \eqref{abs} are not unique, in general, so we focus on the asymptotics of the minimal solution $u_λ^-$ of \eqref{abs}. Under the assumption that constant functions are subsolutions of the critical equation, we prove that the $u_λ^-$ also converges to $u_0$ as $λ\rightarrow 0^-$. Furthermore, we exhibit an example of $H$ for which equation \eqref{abs} admits a unique solution for $λ<0$ as well.
