Existence of C∞ solutions of overdetermined elliptic linear partial differential equations
Project/Area Number  10640188 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Basic analysis

Research Institution  RIKKYO UNIVERSITY 
Principal Investigator 
KAKIE Kunio RIKKYO UNIV. COLLEGE OF SCIENCE, PROFESSOR, 理学部, 教授 (20062664)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)

Keywords  partial differential equations / overdetermined equations / involutiveness / formally integrableness / ellipticity / local differentiable solutions / 線型偏微分方程式系 / 過剰決定系 / 包合系 / 楕円型 / 局所微分可能解 / 局所C^∞解 
Research Abstract 
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 DNeumann 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. 独立変数の個数が3以上のときの考察も行ったが, Spencer列における微分作用素の構造に決定的に異なる様相が生じる為, 方法論と構造に関して手がかかりとなりうる知見を得てはいるが, 問題の一般的解決へ至るには画期的な方法等の新たな発見が必要であり今後の研究課題である.

Report
(4results)
Research Output
(3results)