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

Research Project

Project/Area Number 10640188
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionRIKKYO UNIVERSITY

Principal Investigator

KAKIE Kunio  RIKKYO UNIV. COLLEGE OF SCIENCE, PROFESSOR, 理学部, 教授 (20062664)

Project Period (FY) 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)
Keywordspartial differential equations / overdetermined equations / involutiveness / formally integrableness / ellipticity / local differentiable solutions
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 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.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (3 results)

All Other

All Publications (3 results)

  • [Publications] Kunio Kakie: "Existence of smooth solutions of overdetermined elliptic differential equations in two independent variables"Commentarii Mathematici Universitatis Sancti Pauli. 48. 181-210 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kunio Kakie: "Existence of smooth solutions of overdetermined elliptic differential equations in two independent variables"Commentarii Mathematici Universitatis Sancti Pauli. vol. 48. 181-210 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kunio KAKIE: "Existence of smooth solutions of overdetermined elliptic differential equations in two independent variables"Commentarii Mathematici Universitatis Sancti Pauli. 48・2. 181-210 (1999)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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