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ALGEBRAIC-ANALYTICAL STUDY OF PSEUDO-DIFFERENTIAL AND CONVOLUTION ERUATIONS

Research Project

Project/Area Number 11640153
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

ISHIMURA Ryuichi  FACULTY OF SCIENCES PROFESSOR, 理学部, 教授 (10127970)

Co-Investigator(Kenkyū-buntansha) AOKI Takashi  KINKI UNIVERSITY, FAC.SCIE.TECH.PROFESSOR, 理工学部, 教授 (80159285)
OKADA Yasunori  FACULTY OF SCIENCES ADJOINT PROFESSOR, 理学部, 助教授 (60224028)
HINO Yoshiyuki  FACULTY OF SCIENCES PROFESSOR, 理学部, 教授 (70004405)
TOSE Nobuyuki  KEIO UNIVERSITY, Fac. ECON.PROFESSOR, 経済学部, 教授 (00183492)
TAJIMA Shinichi  NIIGATA UNIVERSITY, Fac. TECH.ADJOINTPROFESSOR, 工学部, 助教授 (70155076)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2000: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsAlgebraic analysis / Convolution equations / Pseudo-differential equations / Partial differential equations / Differential equations of infinite ordre / Micro-local study of sheaves / Canchy problem / 畳込み方程式 / 代数解析
Research Abstract

The aims of this research were as follows :
[1] The algebraic-analytical study of convolution equations in the complex domains, using micro-local study of sheaves.
[2] Study of the Fabry-Ehrenpreis-Kawai Theorem, applying the theory of analytical continuation of solutions to convolution equations.
[3] The extension of the theory of the Cauchy problem for micro-differential equationts in the complex domains to the pseudo-differential case.
For the problem [1] and [2], at first, we constructed good examples of convolution equations with elliptic codition in the several variables. And also, for the problem of the analytic continuation of the holomorphic solutions to the homogeneous convolution equation in the complex domains, we introduced its characteristic sets to be the natural extension of the case of usual constant coefficients linear partial differential equations and we could present the expicit form of the domains to which any solution is continued analytically, using the characteristic set. In particular, this resolves almost completely the problem of the analytic continuation to the infinite ordre differential-difference equations which are important examples of the functional-differential equaion. In the case of tube domains, one proved that, in a natural condition, the characteristic set coincides with the accumulating directions at infinity of the zeros of the symbol. However, for [3], we did not yet succeed to get an general theory. But we are now studying the problem using the sheaf theoritical study by means of inductive limits which is in the course of developpement.

Report

(3 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] R.ISHIMURA: "The characteristic set for differential-difference equations in real domains"kyushu Journal of Mathematics. 53. 1-18 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.ISHIMURA,I.OKPOA and Y.OKADA: "Continuation of holomorphic solutions to Convolution equations in comples domains"Annales Polonici Mathematici. 74. 105-115 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.OKADA: "On the characteristics for convolution equations on tube domains"Journal of Mathematical Society of Japan. 52(3). 535-544 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.HINO,S.MURAKAMI,T.NAITO and N.V.MINN: "A variation-of-constants formula for abstract functional differential equations in the phase space"Journal of Differential Equations. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.OKADA: "On the distribution solutions of micao-differential equations with double involutive characteristics"Communications in PDE. 24(7-8). 1419-1443 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.TAJIMA and Y.NAKAMURA: "Residue calculus with differential operators"Kyushu Journal of Mathematics. 54. 127-138 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] ISHIMURA R.: "The characteristic set for differential-differnce equations in real domains"Kyushu Journal of Mathematics. vol.53. 1-18 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] ISHIMURA R., OKADA J.and OKADA Y.: "Continuatin of holomorphic solutions to convolution equations in complex domains"Annalaes Polonici Mathematici. vol.74. 105-115 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] OKADA Y.: "On the distribution solutions of micro-differential equations with double involutive characteristics"Communications in PDE. vol.24(7-8). 1419-1443 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] OKADA Y.: "On the characteristics for convolution equations on tube domains"Journal of Mathematical Society of Japan. vol.52(3). 535-544 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] HINO Y., MURAKAMI S., NAITO T.and M.V.MINN: "A generalization of processes and stabilities in abstract functional differential equations"Journal of Differential Equations. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] TAJIMA S.and NAKAMURA Y.: "Multidimensional local residues and holonomic D-modules"Kyushu Journal of Mathematics. vol.54. 127-138 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.ISMIMURA,J.OKADA and Y.OKADA: "Continuation of holomorphic solutions to convolution equations in complex domains"Annales Polonici Mathematici. 74. 105-115 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.OKADA: "On the characteristics for convolution equations on tube domains"Journal of Mathematical Society of Japan. 52(3). 535-544 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.HIVO,S.MURAKAMI,J.NAITO and N.V.MINN: "A variation - of - constants formula for abstract functional differential equations in the phase space"Journal of Differential Equations. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] 青木貴史,竹井義次: "Painleve方程式の接続問題とWKB解析"Rokko Lectures in Mathematics. 7. 18-40 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.TAJIMA and Y.NAKAMURA: "Residue calculus with differential operators"Kyushu Journal of Mathematics. 54. 127-138 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.TAJIMA and Y.NAKAMURA: "Computing point residues for a shape basic case via differential operators"京都大学数理解析研究所講究録. 1158. 87-97 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] R. ISHIMURA: "The characteristic set for differential-difference equations in real domains"Kyushu Journal of Mathematics. 53・1. 107-114 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] R. ISHIMURA, J. OKADA and Y. OKADA: "The continuation of holomorphic solutions for convolution equations in complex domains"京都大学数理解析研究所講究録. 1088. 94-100 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] R. ISHIMURA and Y. OKADA: "Examples of convolution equations in tube domains"京都大学数理解析研究所講究録. 1088. 101-105 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y. OKADA: "On distribution solutions of microdifferential equations with double involutive characteristics"Communication in PDE. 24・(7-8). 1419-1443 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T. AOKI, T. KAWAI and Y. TAKEI: "On a complete description of the Stokes geometry for higher order ordinary differential equations with a large parameter via integral representations""Toward the Exact WKB Analysis of Differential Equations, Linear or Non-lineas" Kyoto University Press. 11-14 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] S. Tajima and Y. Nakamura: "Residue calculus with differential operators"Kyushu Journal of Mathematics. 53(to appear). (1999)

    • Related Report
      1999 Annual Research Report

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

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