Existence of C∞ solutions of overdetermined elliptic linear partial differential equations

在过度确定的线性部分差异方程理论中的一个问题可以证明地方差异解决方案的存在。即使在有缺陷的等式的情况下，这个问题也没有得到解决，而没有做出任何评估。在与这个问题的连接中，我们得到了下面的存在理论，是在有两个独立变量的情况下完全解决了这个问题的。一种无形的（或更一般的，正式的）椭圆过度确定的差异等式，有两个独立的变量，可以采用当地无限不同的解决方案，我们提出理论的方式就是遵循。根据一般形式理论，局部存在理论可以被统计为一个对应的短差异复合体的有效性，而更广泛的是等同于对应术语的第二斯宾塞的有效性。为了证明斯宾塞的序列确实低于理论的循环，我们就不能处理D-诺伊曼的问题。Instead介绍斯宾塞序列的运动在L-D12-D1对每个邻居U的感觉，我们展示了这是一个精确的提供U是足够的。在这里，我们可以完全使用斯宾塞序列中不同运算符的事实，有两个独立的变体，它们有简单的地方代表。这一结果与椭圆规律性理论结合在一起，揭示了斯宾塞序列的确切性，以及存在理论的存在性。

One of the problems in the theory of overdetermined linear partial differential equations is to prove the existence of local differentiable solutions. Even in the case of elliptic equations, this problem has not been solved without assuming very strong additional conditions. In connection with this problem, we obtained the following existence theorem, which solves the problem completely in the case of elliptic equations with two independent variables.Theorem. An involutive (or more generally, formally integrable) elliptic overdetermined differential equation with two independent variables admits local infinitely differentiable solutions.The way we prove the theorem is as follows. According to the general formal theory, the local existence theorem may be stated as exactness of a corresponding short differential complex, and the latter is equivalent to exactness of the second Spencer at the corresponding term. To prove that the Spencer sequence is exact under the circumstances of the theorem, we do not treat the D-Neumann problem. Instead introducing the notion of Spencer sequence in LィイD12ィエD1 sense on each neighborhood U, we show that it is exact provided U is small enough. Here we make full use of the fact that the differential operators in the Spencer sequence with two independent variables have simple local representations. This result together with the elliptic regularity theorem implies the exactness of the Spencer sequence, and hence the existence theorem.