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density perturbation

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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Neumann to Steklov eigenvalues: asymptotic and monotonicity results
Part of
Pier Domenico Lamberti, Luigi Provenzano
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 147 / Issue 2 / April 2017

Published online by Cambridge University Press:

16 January 2017, pp. 429-447

Print publication:

April 2017
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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.