J Austral Math Soc Ser A 57 pp305--315, 1994.

The Difference of Consecutive Eigenvalues

Hsu-Tung Ku and Mei-Chin Ku

Abstract

Let M be a smooth bounded domain in Rⁿ with a smooth boundary, n ³ 2, and Du = -åⁿ_i=1¶²u/¶x_i². We prove an inequality involving the first k + 1 eigenvalues of the eigenvalue problem:

ì ï ï ï í ï ï ï î t å m=2  am-1Dmu = lDu in M æ ç è ¶ ¶v ö ÷ ø s   u = 0 on ¶M, s = 0, 1,..., t - 1,

where a_m-1 ³ 0 are constants and a_t-1 = 1. We also obtain a uniform estimate of the upper bound of the ratios of consecutive eigenvalues.

1991 AMS Subject Classification: 35P15

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Authors

Hsu-Tung Ku

Mei-Chin Ku

Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003, USA.