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Spectral stability estimates for the eigenfunctions of second order elliptic operators
Author(s) -
Burenkov Victor I.,
Feleqi Ermal
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201100250
Subject(s) - mathematics , eigenfunction , omega , bounded function , order (exchange) , prime (order theory) , domain (mathematical analysis) , dirichlet boundary condition , homogeneous , differential operator , boundary (topology) , dirichlet distribution , combinatorics , elliptic operator , mathematical analysis , boundary value problem , eigenvalues and eigenvectors , physics , quantum mechanics , finance , economics
Stability of the eigenfunctions of nonnegative selfadjoint second‐order linear elliptic operators subject to homogeneous Dirichlet boundary data under domain perturbation is investigated. Let Ω, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega ^{\prime } \subset \mathbb {R}^n$\end{document} be bounded open sets. The main result gives estimates for the variation of the eigenfunctions under perturbations Ω′ of Ω such that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega _{\varepsilon } = \lbrace x \in \Omega : {\rm dist}(x,\, \mathbb {R}^n \!\setminus \! \Omega ) > \varepsilon \rbrace \subset \Omega ^{\prime } \subset \overline{\Omega ^{\prime }} \subset \Omega$\end{document} in terms of powers of ε, where the parameter ε > 0 is sufficiently small. The estimates obtained here hold under some regularity assumptions on Ω, Ω′. They are obtained by using the notion of a gap between linear operators, which has been recently extended by the authors to differential operators defined on different open sets.