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Second Domain Variation for Problems with Robin Boundary Conditions

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Abstract

In this paper, the first and second domain variations for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions are computed. Extremal properties of the ball among nearly spherical domains of given volume are derived. The discussion leads to a Steklov eigenvalue problem. As a by-product, a general characterization of the optimal shapes is obtained.

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Acknowledgments

The authors would like to thank J. Arrieta, E. Harrell, M. Pierre and the referee for drawing our attention to some of the more recent literature on the subject.

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Authors and Affiliations

  1. Departement Mathematik, Universität Basel, Spiegelgasse 1, 4051, Basel, Switzerland

    Catherine Bandle

  2. Institut für Mathematik, RWTH Aachen, Templergraben 55, 52062, Aachen, Germany

    Alfred Wagner

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  1. Catherine Bandle
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  2. Alfred Wagner
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Correspondence to Alfred Wagner.

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Communicated by Roberto Triggiani.

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Bandle, C., Wagner, A. Second Domain Variation for Problems with Robin Boundary Conditions. J Optim Theory Appl 167, 430–463 (2015). https://doi.org/10.1007/s10957-015-0801-1

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  • DOI: https://doi.org/10.1007/s10957-015-0801-1

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