Asymptotic estimates for the principal eigenvalue of the Laplacian in a geodesic ball
Let M be a compact n-dimensional Riemannian manifold. We consider a point O of M and denote by \(B_{\epsilon}\) the geodesic ball of radius \(\epsilon\) around O and \(M_{\epsilon}=M/\bar B_{\epsilon}\). We consider the Dirichlet problem \(\Delta u=du/dt\) in \(M_{\epsilon}\), \(u|_{dM_{\epsilon}}=0\), \(u|_{t=0}=1_{M_{\epsilon}}\). We denote by \(\lambda_{\epsilon}>0\) the principal eigenvalue of the Laplace-Beltrami operator on \(M_{\epsilon}\). The aim of the paper is to give an explicit asymptotic expansion for \(\lambda_{\epsilon}\) as \(\epsilon_ 0\downarrow\) giving upper and lower bounds. The final result will be that \(\lambda_{\epsilon}\sim C/\log (1/\epsilon)\), \(n=2,\lambda_{\epsilon}\sim Ce^{n-2}\), \(n\geq 2\).
scientific article; zbMATH DE number 3869850