Asymptotics of negative eigenvalues of the Dirichlet problem with the density changing sign

In a three-dimensional anisotropic elastic space with either a bounded foreign inclusion or a void, we derive asymptotic formulas for the increment of the polarization tensor of a defect caused by a smooth variation of the defect boundary. The formulas involve weighted integrals of jumps of the surface enthalpy evaluated for solutions to the problem about deformation of an unperturbed composite space by constant stress at infinity. The study of the positiveness/negativeness of the polarization matrix increment leads to inferences with a clear physical interpretation, in particular, for elastic solids admitting phase transitions. For homogeneous ellipsoid shaped inclusions we derive a relation between the polarization tensor and the Eshelby tensor and obtain miscellaneous consequences of this relation as well. In particular, we introduce the notion of the link tensor which is symmetric and positive definite for any elastic properties of homogeneous materials of the composite space. Bibliography: 60 titles. Illustrations: 5 figures. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 3–36.