标题： 边界曲面上的Hodge-Laplacian特征值 显示英文标题

标题： Hodge-Laplacian Eigenvalues on Surfaces with Boundary

摘要： 最近，Rohleder 提出了一种新的变分方法，用于研究单连通平面上的 Neumann 和 Dirichlet 特征值之间的不等式，使用的是经典向量分析的语言。 用微分形式来表述他的方法，使得这些结果能够推广到更广泛的上下文中。 给出了带边界的黎曼流形上 Hodge-Laplacian 的绝对边界问题的谱，作为闭形式和共余形式空间上绝对边界问题的谱的并集。 通过这种表述，在欧几里得情况下得到了 Hodge-Laplacian 的绝对边界问题和 Laplace-Beltrami 算子的 Dirichlet 边界问题的特征值之间的不等式。 Rohleder 的结果作为更一般定理的推论得到。 显示英文摘要

摘要： Recently Rohleder proposed a new variational approach to an inequality between the Neumann and Dirichlet eigenvalues in the simply connected planar case using the language of classical vector analysis. Writing his approach in terms of differential forms permits to generalize these results to a much broader context. The spectrum of the absolute boundary problem for the Hodge-Laplacian on a Riemannian manifold with boundary is presented as a union of the spectra of the absolute boundary problem on the spaces of closed and co-exact forms. An inequality for the eigenvalues of the absolute boundary problem for the Hodge-Laplacian and the Dirichlet boundary problem for the Laplace-Beltrami operator in the Euclidean case is obtained using this presentation. The Rohleder's results are obtained as corollaries of a more general theorem.