A shape of eigenfunction of the Laplacian under singular variation of domains
Let \(\Omega\) be a bounded domain in \(R^ 3\) with smooth boundary \(\partial \Omega\), and let \(\Omega_{\epsilon}\) be obtained by removing from \(\Omega\) a ball of radius \(\epsilon\) and centre a point w in \(\Omega\). The paper considers the eigenfunctions \(\phi_ j\) and \(\phi_{j,\epsilon}\) in the Dirichlet problems for the Laplacian in \(\Omega\) and \(\Omega_{\epsilon}\). The situation where \(\epsilon\) \(\to 0\) is considered. It is stated that, if \(\phi_ j\) corresponds to a simple eigenvalue, the normal derivative of \(\phi_{j,\epsilon}\) on the boundary of the ball satisfies \(\partial \phi_{j,\epsilon}/\partial n=- \phi_ j(w)\epsilon^{-1}+O(\epsilon^{-1/3}).\) It is also stated that the details of the proof will appear elsewhere.
scientific article; zbMATH DE number 3877656