Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case

Abstract

We obtain the first term in the asymptotic expansion of the eigenvalues of the Laplace operator in a typical dumbbell domain in R 2 {\mathbb {R}^2} . This domain consists of two disjoint domains Ω L {\Omega ^L} , Ω R {\Omega ^R} joined by a channel R ε {R_\varepsilon } of height of the order of the parameter ε \varepsilon . When an eigenvalue approaches an eigenvalue of the Laplacian in Ω L ∪ Ω R {\Omega ^L} \cup {\Omega ^R} , the order of convergence is ε \varepsilon , while if the eigenvalue approaches an eigenvalue which comes from the channel, the order is weaker: ε | ln ε | \varepsilon \left | {{\text {ln}}\varepsilon } \right | . We also obtain estimates on the behavior of the eigenfunctions.