Abstract
We consider an eigenvalue problem for the biharmonic operator with Steklov-type boundary conditions. We obtain it as a limiting Neumann problem for the biharmonic operator in a process of mass concentration at the boundary. We study the dependence of the spectrum upon the domain. We show analyticity of the symmetric functions of the eigenvalues under isovolumetric perturbations and prove that balls are critical points for such functions under measure constraint. Moreover, we show that the ball is a maximizer for the first positive eigenvalue among those domains with a prescribed fixed measure.
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Acknowledgements
The authors are deeply thankful to Prof. Pier Domenico Lamberti who suggested the problem, and also for many useful discussions. The authors acknowledge financial support from the research project ‘Singular perturbation problems for differential operators,’ Progetto di Ateneo of the University of Padova. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Buoso, D., Provenzano, L. (2015). On the Eigenvalues of a Biharmonic Steklov Problem. In: Constanda, C., Kirsch, A. (eds) Integral Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-16727-5_7
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DOI: https://doi.org/10.1007/978-3-319-16727-5_7
Publisher Name: Birkhäuser, Cham
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