Domain-independent upper bounds for eigenvalues of elliptic operators
Domain-independent upper bounds for eigenvalues of elliptic operators
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DOI：
10.1090/s0002-9947-1990-0994167-2
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发表时间：
1990-02
影响因子：
1.3
通讯作者：
S. M. Hook
中科院分区：
文献类型：
--
作者：
S. M. Hook
设qcir ' m是一个有界开集，AQ是它的边界，a是Rm上的拉普拉斯算子。考虑椭圆微分方程：(1)-Au = Au in Q；已知(1)的特征值(i)满足n M n (2) i=l n+1，条件是（n+1 >）。本文抽象了Hile和Protter[2]建立(2)的方法，并将该方法应用于各种二阶椭圆型问题，特别是所有常系数问题。然后，我们考虑了各种高阶问题，并对问题(1)建立了(2)的扩展，其中拉普拉斯算子被Hilbert空间中更一般的算子取代。
Let Q C IR"m be a bounded open set, AQ its boundary and A the Laplacian on Rm . Consider the elliptic differential equation: (1) -Au = Au in Q; u = 0 on AQ. It is known that the eigenvalues, )i, of (1) satisfy n M n (2) i=l n+1l 1 provided that 'n+l > An In this paper we abstract the method used by Hile and Protter [2] to establish (2) and apply the method to a variety of second-order elliptic problems, in particular, to all constant coefficient problems. We then consider a variety of higher-order problems and establish an extension of (2) for problem (1) where the Laplacian is replaced by a more general operator in a Hilbert space.