Ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains (Q2088107)

Let \(\Omega\) be a bounded open subset of \(\mathbb{R}^d\). The authors assume that \(-\Delta\) has a Dirichlet eigenvalue \(\lambda_N(\Omega)\) of multiplicity \(m\geq 1\) in \(\Omega\). Then the authors consider a compact subset \(K\) of \(\Omega\) of capacity equal to \(0\) and a family of compact subsets \(\{K_\epsilon\}_{\epsilon>0}\) that approaches \(K\) in the sense that each neighborhood of \(K\) in \(\Omega\) contains at least an element of the family, and analyze the behavior of the Dirichlet eigenvalues \(\lambda_i(\Omega\setminus K_\epsilon)\) for \(i=N, \dots, N+(m-1)\) of \(-\Delta\) in \(\Omega\setminus K_\epsilon\) that approach \(\lambda_N(\Omega\setminus K)=\lambda_N(\Omega)\) as \(\epsilon\) tends to zero and provide an expansion in terms of extended versions of the capacity of \(K_\epsilon\). Then the authors consider the case \(d=2\) and in which \(K\) consists of a single point (the origin) and provide both asymptotics and series expansions of the extended versions of the capacity in case \(K_\epsilon\) is an \(\epsilon\) multiple of some set of class \(C^{1,\alpha}\), that contains the origin, and exploit their results to prove expansions for the eigenvalues \(\lambda_i(\Omega\setminus K_\epsilon)\) in terms of \(\epsilon\) in such a case.