Approximation of Dirichlet eigenvalues on domains with small holes (Q1900647)

The author derives first order perturbation formulas for the eigenvalues of the Laplacian under Dirichlet boundary conditions in a bounded domain \(\Omega \subset \mathbb{R}^n\), \(n \geq 2\), which contains small holes. The \(H^1_0\)-capacity of the holes essentially plays the role of the perturbation parameter. The author focusses on highly accurate approximation formulas which reflect the relative location of the holes in \(\Omega\) and which are easy to calculate by local quantities. The proofs are based on the minimax principle in combination with two different types of harmonic correction methods for the construction of perturbed eigenfunctions. A couple of concrete examples are given and the paper also contains a review of the history of the field.