Spectral stability estimates for the eigenfunctions of second order elliptic operators
stability estimates for the eigenfunctions
The paper deals with stability of the eigenfunctions of nonnegative selfadjoint second-order linear elliptic operators subject to homogeneous Dirichlet boundary data under domain perturbation. Precisely, let \(\Omega,\Omega'\subset\mathbb R^n\) be bounded and open sets. The main result of the paper provides estimates for the variation of the eigenfunctions under perturbations \(\Omega'\) of \(\Omega\) such that NEWLINE\[NEWLINE \Omega_\varepsilon=\left\{x\in\Omega:\operatorname{dist}(x,\mathbb R^n\setminus\Omega)>\varepsilon\right\}\subset\Omega'\subset\overline{\Omega'} \subset\Omega NEWLINE\]NEWLINE in terms of powers of the small enough parameter \(\varepsilon>0\). These estimates hold under suitable regularity assumptions on \(\Omega\) and \(\Omega'\), and are obtained by means of the notion of gap between linear operators, which has been recently extended by the authors to differential operators defined on different open sets [\textit{V. I. Burenkov} and \textit{E. Feleqi}, Math. Nachr. 286, No.~5--6, 518--535 (2013; Zbl 1268.47055)].
scientific article; zbMATH DE number 6078246