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

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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.