The discrete spectrum of the Laplace operator in a cylinder with locally perturbed boundary (Q1778289)

The author considers a cylindrical domain \(Q_{1}\) in \(\mathbb R^{3}\) obtained by deforming a right cylinder of the form \(\Omega\times\mathbb R\) in a part \(\Omega\times [z_{1},z_{2}]\). Here \(\Omega\) is an open connected subset of \(\mathbb R^{2}\), and \(z_{1},z_{2}\in \mathbb R\). The author proves that under suitable regularity conditions on \(Q_{1}\), the Laplace operator with Dirichlet boundary conditions in \(Q_{1}\) has at least one eigenvalue to which corresponds one eigenfunction in the class \(L^{2}(Q_{1})\cap C^{\infty} (\overline{Q_{1}})\).