New bounds for solutions of second order elliptic partial differential equations
New bounds for solutions of second order elliptic partial differential equations
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DOI：
10.2140/pjm.1958.8.551
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发表时间：
1958-09
影响因子：
0.6
通讯作者：
L. Payne;H. Weinberger
中科院分区：
文献类型：
--
作者：
L. Payne;H. Weinberger
1.导论在前文[10]中，作者给出了确定拉普拉斯方程的Dirichlet积分和解的一点的方法，其精度是任意的和已知的。这些方法具有上下限同时计算的优点。此外，所有的误差估计都是关于任意函数的二次泛函的，因此Rayleigh-Ritz技术给出了一种使误差任意小的系统方法。这些方法依赖于F.Rellich的一个恒等式[12]。因此，必须假定边界相对于某一点是星形的，并且不可能用这些方法来处理变系数的微分方程组。本文将Rellich恒等式推广到一般的二阶椭圆型算子和一大类二阶椭圆型算子组，从而将前文的结果推广到包含这类算子和更一般区域的方程。这个恒等式是由L.H.δrmander[7]得到并用于双曲算子的，他在一份亲切地传达给作者的油印笔记中，独立地从那里获得了一类二阶椭圆型方程解的边值估计。有趣的是注意到恒等式(2.4)的结构与Garabdian和Schiffer给出的关于Ju-Pu的格林函数的第一次变分公式[6]的相似性。为简单起见，本文只详细讨论了不含零阶项的自伴二阶算子的情形。然而，该方法很容易扩展到更一般的运算符，甚至系统，如§6和7所示。第2节涉及上述身份。在§3中，这个恒等式被用来估计N维一般非齐次边值问题的几个重要的二次泛函(包括广义Dirichlet积分)。借助于任意选取的函数的特定泛函，我们得到了广义Dirichlet积分的一个近似。误差估计是任意函数的狄里克莱特数据与给定数据的偏差的二次泛函，并且可以进行
1. Introduction In a previous paper [10] the authors presented methods for determining, with arbitrary and known accuracy, the Dirichlet integral and the value at a point of a solution of Laplace's equation. These methods have the advantage that upper and lower bounds are computed simultaneously. Moreover all error estimates are in terms of quadratic functionals of an arbitrary function, so that the Rayleigh-Ritz technique gives a systematic way of making the error arbitrarily small. These methods depend on an identity of F. Rellich [12]. As a consequence it is necessary to assume that the boundary is star-shaped with respect to some point, and it is not possible to treat differential equations with variable coefficients by these methods. In this paper a generalization of Rellich's identity to general second order elliptic operators as well as to a large class of elliptic systems of second order operators is employed to extend the results of the previous paper to equations involving such operators and rather general domains. The identity in question was obtained and used for hyperbolic operators by L. Hδrmander [7] who, in a mimeographed note kindly communicated to the authors, has independently obtained therefrom some estimates for boundary values of the solution of a second order elliptic equation. It is interesting to note the similarity in structure of the identity (2.4) and the formula for the first variation of Green's function for Ju-pu given by Garabedian and Schiffer [6]. For the sake of simplicity only the case of a self-adjoint second order operator without zero order terms is treated in detail. However the method is easily extended to more general operators, and even systems , as is shown in § §6 and 7. Section 2 is concerned with the above-mentioned identity. In §3 this identity is used to estimate several important quadratic functionals (including the generalized Dirichlet integral) in terms of Dirichlet data for a general non-homogeneous boundary value problem in N dimensions. We obtain an approximation to the generalized Dirichlet integral by means of a specific functional of an arbitrarily chosen function. The error estimate is a quadratic functional in the deviation of the Dirichlet data of the arbitrary function from the given data, and can be made