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Algebra i Analiz, 2016, Volume 28, Issue 2, Pages 34–57 (Mi aa1484)  

This article is cited in 1 scientific paper (total in 1 paper)

Research Papers

Domain perturbations for elliptic problems with Robin boundary conditions of opposite sign

C. Bandlea, A. Wagnerb

a Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland
b Institut für Mathematik, RWTH Aachen, Templergraben 55, D-52062 Aachen, Germany

Abstract: The energy of the torsion problem with Robin boundary conditions is considered in the case where the solution is not a minimizer. Its dependence on the volume of the domain and the surface area of the boundary is discussed. In contrast to the case of positive elasticity constants, the ball does not provide a minimum. For nearly spherical domains and elasticity constants close to zero the energy is the largest for the ball. This result is true for general domains in the plane under an additional condition on the first nontrivial Steklov eigenvalue. For more negative elasticity constants the situation is more involved and is strongly related to the particular domain perturbation. The methods used in this paper are the series representation of the solution in terms of Steklov eigenfunctions, the first and second shape derivatives and an isoperimetric inequality of Payne and Weinberger for the torsional rigidity.

Keywords: Robin boundary condition, energy representation, Steklov eigenfunction, extremal domain, first and second domain variation, optimality conditions.

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English version:
St. Petersburg Mathematical Journal, 2017, 28:2, 153–170

Bibliographic databases:

Received: 30.11.2015
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Citation: C. Bandle, A. Wagner, “Domain perturbations for elliptic problems with Robin boundary conditions of opposite sign”, Algebra i Analiz, 28:2 (2016), 34–57; St. Petersburg Math. J., 28:2 (2017), 153–170

Citation in format AMSBIB
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\by C.~Bandle, A.~Wagner
\paper Domain perturbations for elliptic problems with Robin boundary conditions of opposite sign
\jour Algebra i Analiz
\yr 2016
\vol 28
\issue 2
\pages 34--57
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\transl
\jour St. Petersburg Math. J.
\yr 2017
\vol 28
\issue 2
\pages 153--170
\crossref{https://doi.org/10.1090/spmj/1443}
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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. Bucur D., Fragala I., “On the Honeycomb Conjecture For Robin Laplacian Eigenvalues”, Commun. Contemp. Math., 21:2 (2019), 1850007  crossref  mathscinet  zmath  isi  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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