论文标题
Laplacian的特征值具有移动的混合边界条件：消失的迪里奇地区的情况
Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Dirichlet region
论文作者
论文摘要
在这项工作中，我们考虑了在有限的Lipschitz域上的Laplacian的均匀的Neumann特征值问题，以及它的奇异扰动，该问题包括在边界的一小部分中规定零dirichlet边界条件。我们首先描述了扰动特征值的尖锐渐近行为，在将其收敛到neumann问题的简单特征值的情况下。渐近扩张中的第一项事实取决于子集的Sobolev能力，在该子集中扰动的特征函数正在消失。然后，我们专注于施加在缩放到一个点的子集上的Dirichlet边界条件的情况。通过对电容电位的爆破分析，我们检测到这种收缩的Dirichlet边界部分的Sobolev容量的消失顺序。
In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the boundary. We first describe the sharp asymptotic behaviour of a perturbed eigenvalue, in the case in which it is converging to a simple eigenvalue of the limit Neumann problem. The first term in the asymptotic expansion turns out to depend on the Sobolev capacity of the subset where the perturbed eigenfunction is vanishing. Then we focus on the case of Dirichlet boundary conditions imposed on a subset which is scaling to a point; by a blow-up analysis for the capacitary potentials, we detect the vanishing order of the Sobolev capacity of such shrinking Dirichlet boundary portion.
