SPECTRAL PROBLEMS IN ELASTICITY. SINGULAR BOUNDARY PERTURBATIONS

Jan Sokolowski (1) , Sergeï A. Nazarov (2)

1 IECN - Institut Élie Cartan de Nancy
2 IPME - Institute of Mechanical Engineering Problems [St. Petersburg]

The three-dimensional spectral elasticity problem is studied in an anisotropic and inhomogeneous solid with small defects, i.e., inclusions, voids, and microcracks. Asymptotics of eigenfrequencies and the corresponding elastic eigenmodes are constructed and justified. New technicalities of the asymptotic analysis are related to variable coefficients of differential operators, vectorial setting of the problem, and usage of intrinsic integral characteristics of defects. The asymptotic formulae are developed in a form convenient for application in shape optimization and inverse problems.

Mots clés: Singular perturbations; Spectral problem; Asymptotics of eigenfunctions and eignevalues; Elasticity boundary value problem

Domaines: Equations aux dérivées partielles [math.AP]

Jan Sokolowski, Sergeï A. Nazarov. SPECTRAL PROBLEMS IN ELASTICITY. SINGULAR BOUNDARY PERTURBATIONS. 2010. ⟨hal-00450285⟩