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

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:

Document Type: Article
Received: 30.11.2015
Language: English

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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\paper Domain perturbations for elliptic problems with Robin boundary conditions of opposite sign
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\pages 34--57
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\pages 153--170
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