Lower bound estimates for eigenvalues of the Laplacian

2012/08/26 by Cheng, Qing-Ming, Qi, Xuerong
#35P15 #58C40 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1208.5226

Abstract

For an n-dimensional polytope Ω in ℝⁿ, we study lower bounds for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. In the asymptotic formula on the average of the first k eigenvalues, Li and Yau (1983) obtained the first term with the order k^\frac2n, which is optimal. The next landmark goal is to give the second term with the order k^\frac1n in the asymptotic formula. For this purpose, Kovař'ık, Vugalter and Weidl (2009) have made an important breakthrough in the case of dimension 2. It is our purpose to study the n-dimensional case for arbitrary dimension n. We obtain the second term in the asymptotic sense.

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