Riemann 流形上一类散度形式椭圆微分算子相邻特征值间隙的估计
Estimates of the gaps between consecutive eigenvalues for a class of elliptic differential operators in divergence form on Riemannian manifolds
AI总结：
本文研究 n 维 Euclid 空间有界区域上带 Dirichlet 条件的一类二阶散度形式椭圆算子，建立相邻特征值间隙上界，并涵盖 Laplace 算子与平方 Cheng-Yau 算子及挤压 Cartan-Hadamard 流形特例。
摘要：
在本工作中，我们针对 n 维 Euclid 空间有限区域内、带 Dirichlet 边界条件的一类二阶散度形式椭圆微分算子的特征值问题，得到了相邻特征值之间间隙的上界估计。该算子类包括著名的 Laplace 算子以及平方 Cheng-Yau 算子。在 Laplace 算子情形下，我们的估计与 D. Chen、T. Zheng 和 H. Yang 所得结果一致，而后者在特征值阶数意义下是最优的。对于挤压 Cartan-Hadamard 流形，我们针对该算子的若干特殊情形给出了估计。
英文摘要：
In this work, we obtain estimates for the upper bound of gaps between consecutive eigenvalues for the eigenvalue problem of a class of second-order elliptic differential operators in divergent form, with Dirichlet boundary conditions, in a limited domain of n-dimensional Euclidean space. This class of operators includes the well-known Laplacian and the square Cheng-Yau operator. For the Laplacian case, our estimate coincides with that obtained by D. Chen, T. Zheng, and H. Yang, which is the best possible in terms of the order of the eigenvalues. For pinched Cartan-Hadamard manifolds the estimates were made in particular cases of this operator.
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