The asymptotic form of solutions of singularly perturbed Dirichlet problems

Dirichlet boundary-value problems for the Laplace operator in a plane in domains with thin branches or narrow channels of communication are considered. When the domain is bounded, the method of matched expansions is used to construct power series with respect to a small parameter (the width of the branch) for the eigenvalues, which are identical, at the power level of approximation, with the asymptotic forms of generalized eigenvalues for domains of the Helmholtz resonator type. (C) 1996 Elsevier Science Ltd. All rights reserved.