Neumann to Steklov eigenvalues: asymptotic and monotonicity results

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behaviour of the Neumann eigenvalues and find explicit formulae for their derivatives in the limiting problem. We deduce...

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Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2017-04, Vol.147 (2), p.429-447
Main Authors: Lamberti, Pier Domenico, Provenzano, Luigi
Format: Article
Language:English
Subjects:
Behavior
Boundaries
Boundary conditions
Constraining
Derivatives
Eigenvalues
Formulas (mathematics)
Mathematics
Operators
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Summary:We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behaviour of the Neumann eigenvalues and find explicit formulae for their derivatives in the limiting problem. We deduce that the Neumann eigenvalues have a monotone behaviour in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.
ISSN:0308-2105
1473-7124
DOI:10.1017/S0308210516000214