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

The paper deals with an eigenvalue problems 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 inarXiv.org
Main Authors Bandle, Catherine, Wagner, Alfred
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 15.12.2015
Subjects
Eigenvalues
Inequalities
Resonant frequencies
Shape optimization
Transplantation
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Summary:The paper deals with an eigenvalue problems 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 domain.
ISSN:2331-8422