On the theory of differential boundary problems
On the theory of differential boundary problems
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微分边界问题理论
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发表时间：
1963
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通讯作者：
M. Schechter
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作者：
M. Schechter
在之前的出版物[10]中，作者将负范数方法应用于椭圆边值问题。结果表明，涉及此类范数的不平等如何导致分布边界问题的解决。存在性定理很容易获得，并且具有强大的正则性结果。在本文中，我们提取该方法的基本特征并将其应用于任意偏微分方程的一般边界问题。问题的提出方式包括所有类型的方程和边界条件。我们寻找经典解、强解和弱解存在的充要条件。这些条件通常通过不等式（先验估计）来表达。在应用该理论时，必须证明满足一定的不等式。那么存在就会自动发生。在某些特定情况下（例如，Viik-Sobolev 问题，参见 5），人们发现所需的不等式已经已知，从而立即给出所需的存在定理。我们的主要工具是空间 Ht'(G) 及其子空间上有界线性泛函的表示定理，其中 是任意整数，p 是大于 1 的任意实数，G 是欧几里得 n 空间中的任意域（参见定理 2.1、2.2、6.2、6.3、7.1）。对于 => 0，Ht’’(G) 被定义为 C(C1 G) 相对于范数的完成
In a previous publication [10] the author applied the method of negative norms to elliptic boundary value problems. It was shown how inequalities involving such norms led to solutions of boundary problems for distributions. Existence theorems were easily obtained together with powerful regularity results. In this paper we extract the essential features of the method and apply them to general boundary problems for arbitrary partial differential equations. The problems are posed in such a way as to include ,all types of equations and boundary conditions. We seek necessary and sufficient conditions for the existence of classical, strong, and weak solutions. These conditions are usually expressed by means of inequalities (a priori estimates). In applying the theory one would have to show that a certain inequality is satisfied. Existence then follows automatically. In some particular cases (e.g., the Viik-Sobolev problems, cf. 5) it was discovered that the required inequalities were already known, giving the desired existence theorems immediately. Our main tools are representation theorems for bounded linear functionals on the spaces Ht’(G) and their subspaces, where is an arbitrary integer, p an arbitrary real number greater than one, and G an arbitrary domain in Euclidean n-space (cf. Theorems 2.1, 2.2, 6.2, 6.3, 7.1). For => 0, Ht’’(G) is defined as the completion of C(C1 G) with respect to the norm