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Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 497–512 | Cite as

Extremum problems of Laplacian eigenvalues and generalized Polya conjecture

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Abstract

In this survey on extremum problems of Laplacian-Dirichlet eigenvalues of Euclidian domains, the author briefly presents some relevant classical results and recent progress. The main goal is to describe the well-known conjecture due to Polya, its connections to Weyl’s asymptotic formula for eigenvalues and shape optimizations. Many related open problems and some preliminary results are also discussed.

Keywords

Extremum problems Laplacian eigenvalues Weyl asymptotics Polya’s conjecture Spliting equality Regularity of minimizers 

2000 MR Subject Classification

35 49 57 

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Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew YorkUSA

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