DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

SPECTRAL STABILITY OF ELLIPTIC SELFADJOINT
DIFFERENTIAL OPERATORS WITH DIRICHLET
AND NEUMANN BOUNDARY CONDITIONS
VICTOR I. BURENKOV AND PIER DOMENICO LAMBERTI
We present a general spectral stability theorem for nonnegative selfadjoint operators with
compact resolvents, which is based on the notion of a transition operator, and some applications
to the study of the dependence of the eigenvalues of uniformly elliptic operators
upon domain perturbation.
Copyright © 2006 V. I. Burenkov and P. D. Lamberti. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly
cited.
1. Introduction
Let Ω be a nonempty open set in R N .LetH be a nonnegative selfadjoint operator defined
on a dense subspace of L 2 (Ω) (briefly, a nonnegative selfadjoint operator on L 2 (Ω)) with
compact resolvent. It is well known that the spectrum of H is discrete and its eigenvalues
λ n [H], arranged in nondecreasing order and repeated according to multiplicity, can be
represented by means of the min-max principle. Namely,
λ n [H] =
inf
L⊂Dom(H 1/2 )
dimL=n
sup
u∈L
u≁0
( H 1/2 u,H 1/2 u ) L 2 (Ω)
(u,u) L2 (Ω)
(1.1)
for all n ∈ N,whereH 1/2 denotes the square root of H. (For basic definitions and results,
we refer to Davies [7].)
Here we study the variation of λ n [H] upon variation of H, on the understanding
that Ω may vary as well. Namely, given two nonnegative selfadjoint operators H 1 , H 2
on L 2 (Ω 1 ), L 2 (Ω 2 ), respectively, we aim at finding estimates of the type
λ n
[
H2
]
≤ λn
[
H1
] + cn δ ( H 1 ,H 2
) , (1.2)
where δ(H 1 ,H 2 ) is a prescribed measure of vicinity of H 1 and H 2 ,andc n ≥ 0. To do so, we
present a general spectral stability result which, roughly speaking, claims that the validity
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 237–245