论文标题
严格的诺伊曼问题：渐近专业和“接吻”域
The stiff Neumann problem: asymptotic specialty and "kissing" domains
论文作者
论文摘要
我们研究了在平滑界面的$ω\ subset \ mathbb {r}^d $中，在平滑界面域中为拉普拉斯操作员的僵硬的频谱诺伊曼问题，分为两个子域：一个annulus $ω_1$和核心$ω__0$。密度和刚度常数为$ \ varepsilon^{ - 2m} $和$ \ varepsilon^{ - 1} $ in $ω_0$，而它们是$ 1 $ in $ω_1$的订单$ 1 $。 \ Mathbb {r} $的$ m \ in固定，$ \ varepsilon> 0 $很小。我们为特征值和相应的特征函数提供渐进性，为$ \ varepsilon \至0 $，对于任何$ m $。在尺寸$ 2 $中，当$ω_0$接触外部横向$ \partialΩ$，$ω_1$在一个点$ \ Mathcal {o} $上获得两个尖端时，考虑了两个尖端。使用与“平滑”情况相同的渐近过程的可能性是基于不规则部分附近征征函数的结构。给出了整个渐近系列，以$ x \ to \ mathcal {o} $用于cuspidal域中拉普拉斯操作员的混合边界值问题的解决方案。
We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain $Ω\subset\mathbb{R}^d$ which is divided into two subdomains: an annulus $Ω_1$ and a core $Ω_0$. The density and the stiffness constants are of order $\varepsilon^{-2m}$ and $\varepsilon^{-1}$ in $Ω_0$, while they are of order $1$ in $Ω_1$. Here $m\in\mathbb{R}$ is fixed and $\varepsilon>0$ is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as $\varepsilon \to 0$ for any $m$. In dimension $2$ the case when $Ω_0$ touches the exterior boudary $\partialΩ$ and $Ω_1$ gets two cusps at a point $\mathcal{O}$ is included into consideration. The possibility to apply the same asymptotic procedure as in the "smooth" case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as $x\to\mathcal{O}$ for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given.
