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Neumann to Steklov eigenvalues: asymptotic and monotonicity results

Part of: Partial differential equations Spectral theory and eigenvalue problems Elliptic equations and systems Representations of solutions

Published online by Cambridge University Press:  16 January 2017

Pier Domenico Lamberti  and
Luigi Provenzano
Pier Domenico Lamberti
Affiliation:
Dipartimento di Matematica ‘Tullio Levi-Civita’, Università degli Studi di Padova, Via Trieste 63, 35126 Padova, Italy (lamberti@math.unipd.it; proz@math.unipd.it)
Luigi Provenzano
Affiliation:
Dipartimento di Matematica ‘Tullio Levi-Civita’, Università degli Studi di Padova, Via Trieste 63, 35126 Padova, Italy (lamberti@math.unipd.it; proz@math.unipd.it)
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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.

Keywords

Steklov boundary conditionseigenvaluesdensity perturbationmonotonicityBessel functions

MSC classification

Secondary: 35C20: Asymptotic expansions 35P15: Estimation of eigenvalues, upper and lower bounds 35B25: Singular perturbations 35J25: Boundary value problems for second-order elliptic equations 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 2 , April 2017 , pp. 429 - 447
Copyright
Copyright © Royal Society of Edinburgh 2017