Asymptotics of Dirichlet Eigenvalues and Eigenfunctions of the Laplacian on Thin Domains in R

We consider the Laplace operator with Dirichlet boundary conditions on a domain in R and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This generalizes our previous results in two dimensions and, as in that case, allows us to obtain an approximation for Dirichlet eigenvalues for a large class of domains, under very mild assumptions. As an application, we derive a three–term asymptotic expansion for the first eigenvalue of d−dimensional ellipsoids.