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Estimates for eigenvalues of Laplacian operator with any order

Authors:

Institution:

1. Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China
2. Department of Mathematics, Henan Normal University, Xinxiang 453007, China

Abstract:

Let D be a bounded domain in an n-dimensional Euclidean space ℝⁿ. Assume that

$0 < \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _k \leqslant \cdots$

are the eigenvalues of the Dirichlet Laplacian operator with any order l:

$\left\{ \begin{gathered} ( - \vartriangle )^l u = \lambda u, in D \hfill \\ u = \frac{{\partial u}}{{\partial \vec n}} = \cdots = \frac{{\partial ^{l - 1} u}}{{\partial \vec n^{l - 1} }} = 0, on \partial D \hfill \\ \end{gathered} \right.$

. Then we obtain an upper bound of the (k+1)-th eigenvalue λ_k+1 in terms of the first k eigenvalues.

$\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )} \leqslant \frac{1}{n}4l(n + 2l - 2)]^{\tfrac{1}{2}} \left\{ {\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{1}{2}} \lambda _i^{\tfrac{{l - 1}}{l}} \sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{1}{2}} \lambda _i^{\tfrac{1}{l}} } } } \right\}^{\tfrac{1}{2}}$

. This ineguality is independent of the domain D. Furthermore, for any l ⩾ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang. The first author’s research was supported by the National Natural Science Foundation of China (Grant No. 10571088)

Keywords:

Dirichlet problem eigenvalue estimate Laplacian operator

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