On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems
On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems
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DOI：
10.1002/cpa.3160150203
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发表时间：
1962
影响因子：
3
通讯作者：
S. Agmon
中科院分区：
文献类型：
--
作者：
S. Agmon
本文的目的是得到一般椭圆边值问题特征函数的完备性和特征值的角分布的一般结果。在高阶椭圆算子的Dirichlet问题中，Browder [6，8]得到了具有真实的主部的这类结果。Carleman [10]和Keldys [13]建立了一类非自伴二阶椭圆边值问题特征值的渐近分布和特征函数的完备性的结果。然而，我们注意到，即使对于二阶问题，上述结果也不适用于斜导数边值问题这样一个典型的非自伴问题。我们将考虑一类一般的椭圆边值问题，称之为正则问题。在这个类中，我们将描述一个子类的边界值问题拥有一个离散的频谱。在一些附加条件下，我们将证明（广义）本征函数在各种函数空间中是完备的。对于一个被称为绝对椭圆问题的一般子类，人们可以证明（在算子的主要部分是真实的并适当归一化的情况下）特征值在正轴方向上聚集，并且特征函数是完备的。所有正则二阶椭圆边值问题都是绝对椭圆的。因此，特别是适用于斜导数边值问题前面提到的。在推导我们的结果的主要工具是先验估计椭圆边值问题取决于一个参数。这些估计确保了相关算子的预解式沿着复平面中的某些射线趋于零。这里的基本定理是在第2节中证明的定理2.1。证明利用了[4]中建立的一般L，估计。在第三节中，我们首先将联合收割机定理2.1与关于预解式的其他估计相结合，得到了广义本征函数在
The purpose of this paper is to derive some general results on the completeness of eigenfunctions and the angular distribution of eigenvalues of general elliptic boundary value problems. In the case of the Dirichlet problem for higher order elliptic operators with a real principal part such results were obtained by Browder [6, 8]. Earlier, Carleman [lo] and Keldys [13] established results concerning the asymptotic distribution of eigenvalues and completeness of eigenfunctions of certain non self-ad joint second order elliptic boundary value problems. We note, however, that even for the second order problems the results mentioned do not apply to such a typical non self-adjoint problem as the oblique derivative boundary value problem. We shall consider a general class of elliptic boundary value problems which are termed regular problems. Within this class we shall describe a subclass of boundary value problems possessing a discrete spectrum. With some additional conditions we shall show that the (generalized) eigenfunctions are complete in various function spaces. For a general subclass of problems which are called absolutely elliptic one can show (in case the principal part of the operator is real and properly normalized) that the eigenvalues cluster in the direction of the positive axis and that the eigenfunctions are complete. All regular second order elliptic boundary value problems are absolutely elliptic. Thus in particular the results apply to the oblique derivative boundary value problem mentioned before.The main tool in deriving our results are a priori estimates for elliptic boundary value problems depending on a parameter. These estimates ensure that the resolvent of the associated operator tends to zero along certain rays in the complex plane. The basic theorem here is Theorem 2.1 proved in Section 2. The proof makes use of the general L, estimates established in [4]. In Section 3 we first combine Theorem 2.1 with other estimates on the resolvent to obtain a completeness theorem for generalized eigenfunctions in