L q -PERTURBATIONS OF LEADING COEFFICIENTS OF ELLIPTIC OPERATORS: ASYMPTOTICS OF EIGENVALUES VLADIMIR KOZLOV Received 17 March 2006; Accepted 24 April 2006 We consider eigenvalues of elliptic boundary value problems, written in variational form, when the leading coefficients are perturbed by terms which are small in some integral sense. We obtain asymptotic formulae. The main specific of these formulae is that the leading term is different from that in the corresponding formulae when the perturbation is small in L ∞ -norm. Copyright © 2006 Vladimir Kozlov. 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 Here we consider eigenvalues of boundary value problems for elliptic partial differen- tial equations in variational form with measurable coefficients. The main goal is to de- scribe asymptotics of eigenvalues under small perturbations of coefficients. A specific of the problem is that we suppose that perturbations are small only in some integral sense. Such class of perturbations is quite natural in applications. For example, if coefficients take different values on different parts of the domain and we will study what happened if boundaries between these parts are changed slightly, then we have smallness of the perturbations in L q -norm with q< ∞ but not in L ∞ -norm and we cannot apply in this situation well-known classical results of perturbation theory, see Kato [5], [4, Chapter 8]. Moreover, it appears that even the main term in the asymptotic formula for an eigenvalue is different from the classical one. In order to explain the difference, let us consider the following eigenvalue problem for a symmetric matrix:  λI + B C C ∗ A + D  u v  = μ  u v  , (1.1) where I is the unit matrix, B, A, and D are symmetric matrices. The matrices B, D, and C are considered as small perturbation matrices. Then, as is well known, an approximation Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2006, Article ID 26845, Pages 1–15 DOI 10.1155/AAA/2006/26845