Estimates for eigenvalues of Laplacian operator with any order

作     者：Fa-en WU~(1+) Lin-fen CAO~2 1 Department of Mathematics,Beijing Jiaotong University,Beijing 100044,China 2 Department of Mathematics,Henan Normal University,Xinxiang 453007,China 

作者机构：Department of Mathematics Beijing Jiaotong University Beijing China Department of Mathematics Henan Normal University Xinxiang China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2007年第50卷第8期

页      面：1078-1086页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：the National Natural Science Foundation of China(Grant No.10571088) 

主　　题：Dirichlet problem eigenvalue estimate Laplacian operator 

摘      要：Let D be a bounded domain in an n-dimensional Euclidean space Rn.Assume that 01≤λ2≤…≤λk≤…are the eigenvalues of the Dirichlet Laplacian operator with any order l: （-△）lu=λu,in D u=■=…=■=0,on■■D. Then we obtain an upper bound of the（k+1）-th eigenvalueλk+1in terms of the first k eigenvalues. sum from i=1 to k(λk+1-λi)≤（1/n）[4l（n+2l-2）]1/2{sum from i=1 to k(λk+1-λi)1/2λil-1/lsum from i=1 to k(λk+1-λi)1/2λi1/l}1/2. This ineguaiity is independent of the domain ***,for any l≥3 the above inequality is better than all the known *** rusults are the natural generalization of inequalities corre- sponding to the case l=2 considered by Qing-Ming Cheng and Hong-Cang *** l=1,our inequalities imply a weaker form of Yang *** aslo reprove an implication claimed by Cheng and Yang.