特征值的等周不等式与泛不等式
Isoperimetric and Universal Inequalities for Eigenvalues
摘要：
本文综述了欧几里得空间中有界域上拉普拉斯算子和双拉普拉斯算子的特征值的许多已知不等式。特别地，我们集中于Dirichlet和Neumann Laplacian的低特征值的等周不等式和振动固支板问题（即，具有"Dirichlet“边界条件的双调和算子）。我们还讨论了已知的Dirichlet Laplacian和振动固支板和屈曲问题的特征值的普遍不等式，并提出了一些新的。与这些不等式相关的一些名称是Rayleigh、Faber-Krahn、Saught-Weinberger、Payne-Polya-Weinberger、Sperner、Hile-Protter和H. C.杨偶尔，我们也会评论我们的一些不等式到其他空间中有界域的推广，特别是S^n或H^n。
英文摘要：
This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with ``Dirichlet'' boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to present some new ones. Some of the names associated with these inequalities are Rayleigh, Faber-Krahn, Szego-Weinberger, Payne-Polya-Weinberger, Sperner, Hile-Protter, and H. C. Yang. Occasionally, we will also comment on extensions of some of our inequalities to bounded domains in other spaces, specifically, S^n or H^n.
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