Shape optimization for an elliptic operator with infinitely many positive and negative eigenvalues

This paper deals with an eigenvalue problem possessing infinitely many positive and negative eigenvalues. Inequalities for the smallest positive and the largest negative eigenvalues, which have the same properties as the fundamental frequency, are derived. The main question is whether or not the cla...

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Bibliographic Details
Published in:Advances in nonlinear analysis 2018-02, Vol.7 (1), p.49-66
Main Authors: Bandle, Catherine, Wagner, Alfred
Format: Article
Language:English
Subjects:
15A42
35J20
35N25
49K20
49R05
domain dependence
domain variation
Dynamical boundary condition
eigenvalues
harmonic transplantation
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Summary:This paper deals with an eigenvalue problem possessing infinitely many positive and negative eigenvalues. Inequalities for the smallest positive and the largest negative eigenvalues, which have the same properties as the fundamental frequency, are derived. The main question is whether or not the classical isoperimetric inequalities for the fundamental frequency of membranes hold in this case. The arguments are based on the harmonic transplantation for the global results and the shape derivatives (domain variations) for nearly circular domains.
ISSN:2191-9496
2191-950X
DOI:10.1515/anona-2015-0171