Frontiers in Mathematics

Extremum Problems for Eigenvalues of Elliptic Operators

Authors: Henrot, Antoine

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About this book

Problems linking the shape of a domain or the coefficients of an elliptic operator to the sequence of its eigenvalues are among the most fascinating of mathematical analysis. In this book, we focus on extremal problems. For instance, we look for a domain which minimizes or maximizes a given eigenvalue of the Laplace operator with various boundary conditions and various geometric constraints. We also consider the case of functions of eigenvalues. We investigate similar questions for other elliptic operators, such as the Schrödinger operator, non homogeneous membranes, or the bi-Laplacian, and we look at optimal composites and optimal insulation problems in terms of eigenvalues.

Providing also a self-contained presentation of classical isoperimetric inequalities for eigenvalues and 30 open problems, this book will be useful for pure and applied mathematicians, particularly those interested in partial differential equations, the calculus of variations, differential geometry, or spectral theory.

Reviews

From the reviews:

"The book is a good collection of extremal problems for eigenvalues of elliptic operators and it gives a good account of the present state of research. It presents 30 open problems and is an absolutely necessary starting point for research work in this field. All proofs are strictly rigorous and the author refers for some other proofs to the bibliography, which contains 215 references. The material is interesting for specialists in both pure and applied mathematics, and can also be used in students' work."   —Mathematical Reviews

“This is the first book devoted mainly to this subject and is therefore highly welcome. The book contains many interesting results, documents some recent progress and presents 30 open problems. … The book will help the readers (pure and applied mathematicians interested in this area) to update their knowledge in this lively field of research.” (M. Hoffmann-Ostenhof, Monatshefte für Mathematik, Vol. 159 (3), February, 2010)


Table of contents (11 chapters)

Table of contents (11 chapters)
  • Eigenvalues of elliptic operators

    Pages 1-16

  • Tools

    Pages 17-44

  • The first eigenvalue of the Laplacian-Dirichlet

    Pages 45-60

  • The second eigenvalue of the Laplacian-Dirichlet

    Pages 61-71

  • The other Dirichlet eigenvalues

    Pages 73-83

Buy this book

eBook 44,02 €
price for France (gross)
  • ISBN 978-3-7643-7706-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover 58,01 €
price for France (gross)
  • ISBN 978-3-7643-7705-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Extremum Problems for Eigenvalues of Elliptic Operators
Authors
Series Title
Frontiers in Mathematics
Copyright
2006
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Basel
eBook ISBN
978-3-7643-7706-9
DOI
10.1007/3-7643-7706-2
Softcover ISBN
978-3-7643-7705-2
Series ISSN
1660-8046
Edition Number
1
Number of Pages
X, 202
Topics