RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2009, Volume 21, Issue 5, Pages 155–195 (Mi aa1157)

Research Papers

The Eshelby theorem and the problem on optimal patch

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

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.

Full text: PDF file (435 kB)
References: PDF file   HTML file

English version:
St. Petersburg Mathematical Journal, 2010, 21:5, 791–818

Bibliographic databases:

MSC: 35J57, 74B05

Citation: S. A. Nazarov, “The Eshelby theorem and the problem on optimal patch”, Algebra i Analiz, 21:5 (2009), 155–195; St. Petersburg Math. J., 21:5 (2010), 791–818

Citation in format AMSBIB
\Bibitem{Naz09} \by S.~A.~Nazarov \paper The Eshelby theorem and the problem on optimal patch \jour Algebra i Analiz \yr 2009 \vol 21 \issue 5 \pages 155--195 \mathnet{http://mi.mathnet.ru/aa1157} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2604567} \zmath{https://zbmath.org/?q=an:1204.35085} \transl \jour St. Petersburg Math. J. \yr 2010 \vol 21 \issue 5 \pages 791--818 \crossref{https://doi.org/10.1090/S1061-0022-2010-01118-X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000282186800008} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84861627507} 

• http://mi.mathnet.ru/eng/aa1157
• http://mi.mathnet.ru/eng/aa/v21/i5/p155

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Leugering G. Nazarov S. Schury F. Stingl M., “The Eshelby theorem and application to the optimization of an elastic patch”, SIAM J. Appl. Math., 72:2 (2012), 512–534
2. Schneider M., Andrae H., “The Topological Gradient in Anisotropic Elasticity With An Eye Towards Lightweight Design”, Math. Meth. Appl. Sci., 37:11 (2014), 1624–1641
3. Gryshchuk S., de Cristoforis M.L., “Simple Eigenvalues For the Steklov Problem in a Domain With a Small Hole. a Functional Analytic Approach”, Math. Meth. Appl. Sci., 37:12 (2014), 1755–1771
4. Leugering G., Nazarov S.A., “The Eshelby Theorem and Its Variants For Piezoelectric Media”, Arch. Ration. Mech. Anal., 215:3 (2015), 707–739
5. Schury F., Greifenstein J., Leugering G., Stingl M., “on the Efficient Solution of a Patch Problem With Multiple Elliptic Inclusions”, Optim. Eng., 16:1 (2015), 225–246
6. Freidin A.B., Kucher V.A., “Solvability of the Equivalent Inclusion Problem For An Ellipsoidal Inhomogeneity”, Math. Mech. Solids, 21:2, SI (2016), 255–262
7. Novotny A.A., Sokolowski J., Zochowski A., “Topological Derivatives of Shape Functionals. Part i: Theory in Singularly Perturbed Geometrical Domains”, J. Optim. Theory Appl., 180:2 (2019), 341–373
•  Number of views: This page: 586 Full text: 139 References: 37 First page: 16