Extremum Problems of Laplacian Eigenvalues and Generalized Polya Conjecture(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

摘要

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.