Universal inequalities for the eigenvalues of a power of the Laplace operator
Universal inequalities for the eigenvalues of a power of the Laplace operator
复制标题
DOI：
10.1007/s00229-010-0338-4
复制
发表时间：
2010-01
影响因子：
0.6
通讯作者：
S. Ilias;Ola Makhoul
中科院分区：
文献类型：
--
作者：
S. Ilias;Ola Makhoul
本文得到了一个新的抽象公式，它将自伴算子的特征值与两类对称和反对称算子及其算子联系起来。这个公式推广了Harrell、Stubbe、Hook、Ashbaugh、Hermi、Levitin和Parnovski等人的早期公式。我们还展示了如何可以使用这个抽象的配方都给予不同的和更简单的证明所有已知的结果得到的权力的拉普拉斯算子的特征值（即狄利克雷拉普拉斯算子，夹板问题的bilaplacian和更一般的多调和问题上的有界欧氏域），并获得新的。在最后一段中，我们推导出了海森堡群上科恩-拉普拉斯算子的任意幂的特征值的新界。
In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use this abstract formulation both for giving different and simpler proofs for all the known results obtained for the eigenvalues of a power of the Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for the bilaplacian and more generally for the polyharmonic problem on a bounded Euclidean domain) and to obtain new ones. In a last paragraph, we derive new bounds for eigenvalues of any power of the Kohn Laplacian on the Heisenberg group.