Extrema of low eigenvalues of the Dirichlet-Neumann Laplacian on a disk

Details Extrema of low eigenvalues of the Dirichlet-Neumann Laplacian on a disk Extrema of low eigenvalues of the Dirichlet-Neumann Laplacian on a disk

2007-08-26

arxiv.org/abs/0708.3489v1

Canadian Journal of Mathematics 62 (2010), pp. 808-826

Mathematics

Spectral Theory

13 pages, 6 figures

Scientific paper

10.4153/CJM-2010-042-8

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.

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