Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case

José M. Arrieta
1995 Transactions of the American Mathematical Society  
We obtain the first term in the asymptotic expansion of the eigenvalues of the Laplace operator in a typical dumbbell domain in E2 . This domain consists of two disjoint domains Í2L, iV* joined by a channel Re of height of the order of the parameter e . When an eigenvalue approaches an eigenvalue of the Laplacian in ClL uClR , the order of convergence is £ , while if the eigenvalue approaches an eigenvalue which comes from the channel, the order is weaker: e| lne| . We also obtain estimates on
more » ... btain estimates on the behavior of the eigenfunctions.
doi:10.1090/s0002-9947-1995-1297521-1 fatcat:nlyuclgszjg7xbqu4brhmqo3km