Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. As an application, we prove that, again under an explicit nondegeneracy condition, the case of resonant domains features polynomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions.
Abatangelo, L., Felli, V., Terracini, S. (2014). On the sharp effect of attaching a thin handle on the spectral rate of convergence. JOURNAL OF FUNCTIONAL ANALYSIS, 266(6), 3632-3684 [10.1016/j.jfa.2013.11.019].
On the sharp effect of attaching a thin handle on the spectral rate of convergence
ABATANGELO, LAURA;FELLI, VERONICA;
2014
Abstract
Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. As an application, we prove that, again under an explicit nondegeneracy condition, the case of resonant domains features polynomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.