论文标题
关于周期性汉密尔顿的紧凑型扰动的消失折扣近似：1d案例
On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case
论文作者
论文摘要
我们研究粘度解决方案的渐近行为$ u^λ_g$ hamilton-jacobi（hj）方程\ begin \ begin {等式*}λu（x）+g（x，x，x，u'）= c（g）其中$ g（x，p）：= h（x，p）-v（x）$是c（{{\ mathbb r} \ times {\ mathbb r}）$，$ {\ mathbb z} $的扰动{C} _C（{\ Mathbb r}）$。上面出现的常数$ c（g）$定义为{\ mathbb r} $ in {\ mathbb r} $的最小值$ a \ hj方程$ g（x，x，u'）= a $ in $ {\ mathbb r} $ in $ {\ mathbb r} $承认有限的粘度子分子。我们证明了功能$ u^λ_g$本地均匀收敛，$λ\ rightArrow 0^+$，到一个特定的解决方案$ u_g^0 $的critical Quequation \ begin \ begin {equication} \ label {abs} {abs} {abs} \ tag \ tag \ tag {*} \ end {equation}我们在预计的$ g $的Mather量度和限制$ u^0_h $的预计Mather度量中识别$ u^0_g $。这可以被视为[17]中主要结果的非恰当设置的扩展。我们的工作还包括对\ eqref {abs}的定性分析，具有弱的kam理论风味。
We study the asymptotic behavior of the viscosity solutions $u^λ_G$ of the Hamilton-Jacobi (HJ) equation \begin{equation*} λu(x)+G(x,u')=c(G)\qquad\hbox{in $\mathbb{R}$} \end{equation*} as the positive discount factor $λ$ 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 $u^λ_G$ locally uniformly converge, for $λ\rightarrow 0^+$, 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.
