Journal of Evolution Equations

, Volume 3, Issue 3, pp 443–461 | Cite as

On the motion of rigid bodies in a viscous incompressible fluid

Dedicated to the memory of Philippe Benilan

Abstract.

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.

Eigenvalues minimization isoperimetric inequalities optimal domain 

Copyright information

© Birkhäuser-Verlag Basel 2003

Authors and Affiliations

  1. 1.Ecole des Mines and Institut Elie Cartan NancyUMR 7502 CNRS and Projet Corida INRIAVandœuvre-lès-Nancy

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