论文标题
多谐波运算符在小孔的域中的扰动特征值
Perturbed eigenvalues of polyharmonic operators in domains with small holes
论文作者
论文摘要
我们研究了在切除小型内部紧凑型集合中，多谐操作员在有限域中的特征值的奇异扰动。我们在外部边界上考虑均匀的dirichlet和navier条件，同时我们将均匀的dirichlet条件施加在删除集的边界上。为此，我们制定了适合我们的高阶环境的能力概念，并允许根据删除集合的能力来获取扰动的简单特征值的渐近性尤吉特值的描述。然后，在缩放到一个点的子集的特定情况下，我们应用爆破分析来检测精确的收敛速率，事实证明这取决于本征功能的消失顺序。在这方面，Hardy-Rellich的不平等现象是为了确定包含限制概况的适当功能空间。值得注意的是，无论外部边界上规定的边界条件如何，这都是相同的。
We study singular perturbations of eigenvalues of the polyharmonic operator on bounded domains under removal of small interior compact sets. We consider both homogeneous Dirichlet and Navier conditions on the external boundary, while we impose homogeneous Dirichlet conditions on the boundary of the removed set. To this aim, we develop a notion of capacity which is suitable for our higher-order context, and which permits to obtain a description of the asymptotic behaviour of perturbed simple eigenvalues in terms of a capacity of the removed set, in dependence of the respective normalized eigenfunction. Then, in the particular case of a subset which is scaling to a point, we apply a blow-up analysis to detect the precise convergence rate, which turns out to depend on the order of vanishing of the eigenfunction. In this respect, an important role is played by Hardy-Rellich inequalities in order to identify the appropriate functional space containing the limiting profile. Remarkably, for the biharmonic operator this turns out to be the same, regardless of the boundary conditions prescribed on the exterior boundary.
