Inequalities for lower order eigenvalues of second order elliptic operators in divergence form on Riemannian manifolds
Inequalities for lower order eigenvalues of second order elliptic operators in divergence form on Riemannian manifolds
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DOI：
10.1007/s00013-013-0564-6
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发表时间：
2013-10
影响因子：
0.6
通讯作者：
He-Jun Sun;Daguang Chen
中科院分区：
文献类型：
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作者：
He-Jun Sun;Daguang Chen
本文研究完备黎曼流形的有界域上散度形式的二阶椭圆算子的Dirichlet特征值问题。我们讨论了欧氏空间中的子流形、允许球面特征映射的黎曼流形以及允许函数和Δfα=0的黎曼流形，其中∇是梯度算子。建立了这类问题低阶特征值的几个不等式。作为这些结果的应用，我们得到了Dirichlet Laplace问题低阶特征值的一些普遍的不等式。特别地，单位球面上拉普拉斯算子的特征值的泛函不等式是最优的。
In this paper, we investigate the Dirichlet eigenvalue problems of second order elliptic operators in divergence form on bounded domains of complete Riemannian manifolds. We discuss the cases of submanifolds immersed in a Euclidean space, Riemannian manifolds admitting spherical eigenmaps, and Riemannian manifolds which admitlfunctionssuch thatand Δfα= 0, where ∇ is the gradient operator. Some inequalities for lower order eigenvalues of these problems are established. As applications of these results, we obtain some universal inequalities for lower order eigenvalues of the Dirichlet Laplacian problem. In particular, the universal inequality for eigenvalues of the Laplacian on a unit sphere is optimal.