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Extrema of Low Eigenvalues of the Dirichlet–Neumann Laplacian on a Disk

Published online by Cambridge University Press:  20 November 2018

Eveline Legendre
Eveline Legendre*
Affiliation:
Université de Montréal, Montréal, QC H3C 3J7, e-mail: egendre@dms.umontreal.ca
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Abstract

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We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

Keywords

35J2535P15LaplacianeigenvaluesDirichlet–Neumann mixed boundary conditionZaremba’s problem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 4 , 01 August 2010 , pp. 808 - 826
Copyright
Copyright © Canadian Mathematical Society 2010

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