标题： 用de Rham复形改进Friedlander-Filonov不等式 显示英文标题

标题： Refining Friedlander-Filonov inequality with the de Rham complex

摘要： 在过去的几十年里，对拉普拉斯算子和其他微分算子的狄利克雷和诺伊曼特征值之间的不等式进行了深入研究。本文的目的是引入微分形式和德·拉姆复形，以研究此类不等式。我们展示了微分形式如何隐藏在罗赫勒关于平面区域上拉普拉斯算子的狄利克雷和诺伊曼特征值之间不等式工作的核心之中。此外，我们通过证明对于二维或更高维的紧致利普希茨域，诺伊曼拉普拉斯算子的j+2阶特征值小于或等于狄利克雷拉普拉斯算子的j阶特征值，从而扩展和推广了罗赫勒的结果。 显示英文摘要

摘要： Inequalities between Dirichlet and Neumann eigenvalues of the Laplacian and of other differential operators have been intensively studied in the past decades. The aim of this paper is to introduce differential forms and the de Rham complex in the study of such inequalities. We show how differential forms lie hidden at the heart of work of Rohleder on inequalities between Dirichlet and Neumann eigenvalues for the Laplacian on planar domains. Moreover, we extend and generalize the result of Rohleder by proving that the j+2:nd eigenvalue of the Neumann Laplacian is less than or equal to the j:th eigenvalue of the Dirichlet Laplacian for any compact Lipschitz domain in dimension two or greater.