Viewing the Steklov Eigenvalues of the Laplace Operator as Critical Neumann Eigenvalues

Part of the Trends in Mathematics book series (TM)

Abstract

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of boundary mass concentration. We discuss the asymptotic behavior of the Neumann eigenvalues in a ball and we deduce that the Steklov eigenvalues minimize the Neumann eigenvalues. Moreover, we study the dependence of the eigenvalues of the Steklov problem upon perturbation of the mass density and show that the Steklov eigenvalues violates a maximum principle in spectral optimization problems.

Keywords

Steklov boundary conditions Eigenvalues Optimization 

Mathematics Subject Classification (2010)

35J25 35B25 35P15 

Notes

Acknowledgements

We acknowledge financial support by the research project “Singular perturbation problems for differential operators”, Progetto di Ateneo of the University of Padova.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di PadovaPadovaItaly

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