Abstract
We consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementary symmetric functions of multiple eigenvalues are real analytic, and provide Hadamard-type formulas for the corresponding shape derivatives. After recalling the known results in shape optimization, we prove that balls are always critical domains under volume constraint.
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Acknowledgments
The author is very thankful to Prof. Pier Domenico Lamberti and Dr. Luigi Provenzano for useful comments and discussions. The author has been partially supported by the research project ‘Singular perturbation problems for differential operators’ Progetto di Ateneo of the University of Padova, and by the research project FIR (Futuro in Ricerca) 2013 ‘Geometrical and qualitative aspects of PDE’s’. The author is a member 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. (2016). Analyticity and Criticality Results for the Eigenvalues of the Biharmonic Operator. In: Gazzola, F., Ishige, K., Nitsch, C., Salani, P. (eds) Geometric Properties for Parabolic and Elliptic PDE's. GPPEPDEs 2015. Springer Proceedings in Mathematics & Statistics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-41538-3_5
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DOI: https://doi.org/10.1007/978-3-319-41538-3_5
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