Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces
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Abstract
We prove two explicit bounds for the multiplicities of Steklov eigenvalues <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given Riemannian surface with smooth boundary the multiplicities of Steklov eigenvalues <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> are uniformly bounded in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> .
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