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

This article is cited in 7 scientific papers (total in 7 papers)

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.

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

Bibliographic databases:

MSC: 35J57, 74B05
Received: 24.03.2009

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
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    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    4. Leugering G., Nazarov S.A., “The Eshelby Theorem and Its Variants For Piezoelectric Media”, Arch. Ration. Mech. Anal., 215:3 (2015), 707–739  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  scopus
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