Extremum problems of Laplacian eigenvalues and generalized Polya conjecture

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

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