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On mixed Dirichlet-Neumann eigenvalues of triangles

Author: Bartłomiej Siudeja
Journal: Proc. Amer. Math. Soc. 144 (2016), 2479-2493
MSC (2010): Primary 35P15
Published electronically: October 14, 2015
MathSciNet review: 3477064
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Abstract: We order the lowest mixed Dirichlet-Neumann eigenvalues of right triangles according to which sides we apply the Dirichlet conditions. It is generally true that the Dirichlet condition on a superset leads to larger eigenvalues, but it is nontrivial to compare e.g. the mixed cases on triangles with just one Dirichlet side. As a corollary we also classify the lowest Neumann and Dirichlet eigenvalues of rhombi according to their symmetry/antisymmetry with respect to the diagonals.

Furthermore, we give an order for the mixed Dirichlet-Neumann eigenvalues on arbitrary triangle, assuming two Dirichlet sides. The single Dirichlet side case is conjectured to also have the appropriate order, following the right triangular case.

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Bartłomiej Siudeja
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Keywords: Variational bounds, symmetrization, rhombi, Zaremba problem
Received by editor(s): February 12, 2015
Received by editor(s) in revised form: July 7, 2015
Published electronically: October 14, 2015
Communicated by: Michael Hitrik
Article copyright: © Copyright 2015 American Mathematical Society

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