Neumann to Steklov eigenvalues: asymptotic and monotonicity results

Abstract

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 behavior of the Neumann eigenvalues and find explicit formulas for their derivatives at the limiting problem. We deduce that the Neumann eigenvalues have a monotone behavior in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.Comment: This paper has been accepted for publication in Proceedings of the Royal Society of Edinburgh Section A Mathematics and will appear in a revised form subsequent to editorial input by the ICMS/Royal Soc. of Edinburgh. Material on these pages is copyright Cambridge University Press. http://www.rsescotlandfoundation.org.uk/proceedings-a-mathematics.html journals.cambridge.org/action/displayJournal?jid=PR