2014/11/05 by Yue He, He, Yue
Mathematics · #35B50 #35J05 #58C40 #58J05 #65N25 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications #Primary 35P15 #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1411.1135
openalex publication_date 2014/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain Ω in ℝn. It is well known that the k-th Dirichlet eigenvalue λk obeys the Weyl asymptotic formula, that is, λk∼(4π2)/((ωnvolΩ)^(2)/(n))k^(2)/(n) \hboxas k→∞, where volΩ is the volume of Ω. In view of the above formula, Pólya conjectured that λk\gs(4π2)/((ωnvolΩ)^(2)/(n))k^(2)/(n) \hboxfor k∈ℕ. This is the well-known conjecture of Pólya. Studies on this topic have a long history with much work.In particular, one of the more remarkable achievements in recent tens years has been achieved by Li and Yau [Comm. Math. Phys. 88 (1983), 309--318]. They solved partially the conjecture of Pólya with a slight difference by a factor n/(n+2). Here, following the argument of Li and Yau on the whole, we shall thoroughly solve the above conjecture.