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PSEUDO-DIFFERENTIAL EQUATIONS IN COMPLEX DOMAINS

Research Project

Project/Area Number 09640155
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionCHIBA UNIVERSITY

Principal Investigator

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

Co-Investigator(Kenkyū-buntansha) TOSE Nobuyuki  KEIO UNIVERSITY,Fac.ECON., PROFESSOR, 経済学部, 教授 (00183492)
TAJIMA Shinichi  NIIGATA UNIVERSITY,Fac.TECH., A DJOINT PROFESSO, 工学部, 助教授 (70155076)
AOKI Takashi  KINKI UNIVERSITY,FAC.SCI.&TECH., A DJOINT PROFESSOR, 理工学部, 助教授 (80159285)
OKADA Yasunori  FACULTY OF SCIENCES,A DJOINT PROFESSOR, 理学部, 助教授 (60224028)
HINO Yoshiyuki  FACULTY OF SIENCES,PROFESSOR, 理学部, 教授 (70004405)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1997: ¥1,600,000 (Direct Cost: ¥1,600,000)
KeywordsAlgebraic analysis / pseudo differential equations / convolution equations / differential equations of infinite order / parcial differential equations / differential-difference equations / Micro-local analysis / Analytic Cortinuation / 合成積方程式 / 微分・差分方程式 / 非特性コ-シ-問題 / 層の超局所理論
Research Abstract

The aims of this research were as follows :
[1] The non-charasteristic Cauchy problem of the system of micro-differential equations for holomporphic functions.
[2] The fundemental principle for the systems of (pseudo-)differential equations of infinite order.
For the problem [1], using the action of pseudo-differential operators established by M.Kashiwara and P.Schapira, we proved the Cauchy-Kowalevskaya theorem at a micro-local direction p as the isomorphism in the derived category D_b (X ; p). By this grant, we invited Professor P.Schapira of Universite Paris VI to have discussions and to be sugested. In fact, it was quite important for the research, his sugestions and many discussions with him. For the problem [2], we studied the continuation of holomorphic solutions for convolution equations in the complex domains. We defined the characteristic set of the operator as a natural generalization of the case of differential operators and using this notion, we proved the analytic continuation of solutions to any direction determined by the characteristic set. We applied this theory to the differential-difference equation case and we gave explicitly its characteristic set. We constructed also an example of the convolution equations having all good properties.
In this research, we also obtained many results concerning the study of stabilities and the existence of almost periodic solutions in abstract functional differential equations, the study of the 2-micro-hyperbolic pseudodifferential equations, the study of determination of Stokes geometry of third order ordinary differential equations having large parameters, the study of Grothendieck residue and the algorithm computing the duality.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (25 results)

All Other

All Publications (25 results)

  • [Publications] R. ISHIMURA: "The Cauchy-Kowalevski theorem for ε-modules" Journal de Mathematiques Pures et Appliguees. 77. 647-654 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] R. ISHIMURA: "The characteristic set for differential-difference equations in real domains." Kyushu Journal of Mathematics. 53(to appear). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y. HINO and S. MURAKAMI: "A generalization of processes and stabilities in abstract functional differential equations" Funkcialaj Ekvacioj. 41. 235-253 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] O. LIESSE, Y. OKADA and N. TOSE: "A remark on 2-microhyperbolicity" Proceedings of the Japan Academy. 74(A). 39-42 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T. AOKI, T. KAWAI and Y. TAKEI: "On the exact WKB analysis for the third order ordinary differential equations with a large parameter" Asian Journal of Mathematics. 2-4(to appear). (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] S. TAJIMA, T. OAKU and Y. NAKAMURA: "Multidimensional local residues and holonomic D-modules" 京都大学数理解析研究所講究録. 1033. 59-70 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] ISHIMURA R.: "The Cauchy-Kowalevski theorem for E-modules" Journal de Ma thematiques Pures et Appliquees. vol.77. 647-654 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] ISHIMURA R.: "The characteristic set for differential-differnce equations in real domains" Kyushu Journal of Mathematics. vol.53(to appear). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] HINO Y.and MURAKAMI S.: "A generalization of processes and stabilities in abstract functional differential equations" Funkcialaj Ekvacioj. vol.41. 235-255 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] LIESSE O., OKADA Y.and TOSE N.: "A remark on 2-microhyper-bolicity" Proceedings of the Japan Academy. vol.74(A). 39-42 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] AOKI T., KAWAI T.and TAKEI Y.: "On the exact WKB analysis for the third order ordinary differential equations with a large parameter" Asian Journal of Mathematics. vol.2, No.4(to appear). (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] TAJIMA S., OAKU T.and NAKAMURA Y.: "Multidimensional local residues and holonomic D-modules" Suuriken Kakyuroku. vol.1033. 59-70 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] R.ISHIMURA: "The Cauchy-Kowalevski theorem for 2-modules" Journal de Mathematiques Pures et Appliquees. 77. 647-654 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] R.ISHIMURA: "The characteristic set for differential-difference equations in real domains" Kyushu Journal of Mathematics. 53 (to aprear). 1-18 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.HINO and S.MURAKAMI: "A generalization of processes and stabilities in abstract fanctional differential equations" Funkcialaj Ekracioj. 41. 235-255 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] O.LIESSE,Y.OKADA and N.TOSE: "A remark on 2-microhyperbolicity" Proceedings of the Japan Academy. 74(A). 39-42 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.AOKI,T.KAWAI and Y.TAKEI: "On the exact WKB analysrs for the thirdorder ordirary differential equations with a large parameter" Asian Journal of Mathematics. 2-4 (to appear). (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] S.TAJIMA,T.OAKU and Y.NAKAMURA: "Multidimensional local residues and holonomic D-modules" 京都大学数理解析研究所講究録. 1033. 59-70 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] R.ISHiMuRA and J.OKADA: "The continuation of solutions for convolution esuations in complex domain" 京都大学数理解析研究所講究録「Resurgent Functions と合成積方程式」. (予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.OKADA: "Solvability and propagation of singularities for some class of microdifferential eguations in the spaces of micro-distributions" Proceedings of the Seventh International Colloquium on Differential Equations,Plovdiv,Bulgaria. 289-294 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.AoKi: "Instanton-type formal solutions to the second Painleve equations with a large parameter" New Trends in Microlocal Aralysis,Springer. 103-112 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.AoKi: "Multiple-scale analysis for Painleve transcendents with a large parameter" Banach Conter Publications,Polish Academy. 39. 11-17 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] S.Tajima: "Grothen dieck residue calcalus and holonomic D-modules" Proceedings of the Fifth Interhational Conference on Complex Analysis,Beijing (in press). (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] S.Tajima: "Geometric Phases and Bloch Electrons" JSPS-DOST,Lecture Wotes in Mathematics. 5. (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 日野義之・石村隆一・久我健一: "基礎からの微分積分" 培風館, 150 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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