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The Eshelby theorem and patch optimization problem


Author: S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 21 (2009), nomer 5.
Journal: St. Petersburg Math. J. 21 (2010), 791-818
MSC (2010): Primary 35J57, 74B05
DOI: https://doi.org/10.1090/S1061-0022-2010-01118-X
Published electronically: July 15, 2010
MathSciNet review: 2604567
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Abstract: Let $ \Omega^0$ be an ellipsoidal inclusion in the Euclidean space $ {\mathbb{R}}^n$. It is checked that if a solution of the homogeneous transmission problem for a formally selfadjoint elliptic system of second order differential equations with piecewise smooth coefficients grows linearly at infinity, then this solution is a linear vector-valued function in the interior of $ \Omega^0$. This fact generalizes the classical Eshelby theorem in elasticity theory and makes it possible to indicate simple and explicit formulas for the polarization matrix of the inclusion in the composite space, as well as to solve a problem about optimal patching of an elliptical hole.


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Additional Information

S. A. Nazarov
Affiliation: Institute of Mechanical Engineering Problems, Bol’shoi Prospekt V.O. 61, St.Petersburg 199178, Russia
Email: srgnazarov@yahoo.co.uk

DOI: https://doi.org/10.1090/S1061-0022-2010-01118-X
Keywords: Formally selfadjoint elliptic system, junction conditions, ellipsoidal inclusion, Eshelby theorem, optimization of inclusion
Received by editor(s): March 24, 2009
Published electronically: July 15, 2010
Additional Notes: Supported by RFBR (grant no. 09-01-00759)
Article copyright: © Copyright 2010 American Mathematical Society