Laplace算子特征值和的精细下界

Refined Lower Bound for Sums of Eigenvalues of the Laplace Operator

该文研究了R^(n)中Laplace算子在有界域Ω上的Dirichlet特征值和的下界.众所周知:第k个Dirichlet特征值λk(Ω)服从Weyl渐近公式,即λk(Ω)~4π^(2)/[wnV(Ω)]^(2)/nk^(2)/n当k→∞时,其中wn和V(Ω)分别为是R^(n)中n维单位球的体积和Ω的体积.根据上述公式,Pólya猜测λk(Ω)≥4π2/[wnV(Ω)]2/nk^(2)/n,■k∈N.这就是著名的Pólya猜想.对这一问题的研究由来已久,已有很多的工作.特别是,近几十年来最显著的成就之一是由Berezin[4],以及李伟光和丘成桐[3]分别独立取得的.他们部分解决了Pólya猜想,只是多了一个因子n/(n+2).后来,Melas^([7])改进了Berezin-Li-Yau的估计,在不等式右边增加了一个正的k阶项.该文采用与Melas几乎相同的论证,进一步完善了Melas的估计.

In this paper,we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domainΩin R^(n).It is well known that the k-th Dirichlet eigenvalueλk(Ω)obeys the Weyl asymptotic formula,that is,λk(Ω)~4π~2/[wnV(Ω)]2/nk^(2)/nas k→∞,where wnand V(Ω)are the volume of n-dimensional unit ball in R^(n)and the volume ofΩrespectively.In view of the above formula,Polya conjectured thatλk(Ω)≥4π^(2)/[wnV(Ω)]2/nk^(2)/nfor k∈N.This is the well-known conjecture of Polya.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 independently by Berezin[2]and Li and Yau[4],respectively.They solved partially the conjecture of Pólya with a slight difference by a factor n/(n+2).Later,Melas^([7])improved Berezin-Li-Yau’s estimate by adding an additional positive term of the order of k to the right side.Here,following almost the same argument as Melas,we further refine Melas’s estimate.