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Modern analysis of fundamental structures of solutions to partial differential equations

Research Project

Project/Area Number 10640164
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

OKAJI Takashi  Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20160426)

Co-Investigator(Kenkyū-buntansha) SHIGEKAWA Ichiro  Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00127234)
NISHIDA Takaaki  Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70026110)
IKAWA Mitsuru  Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80028191)
DOI Shin'ichi  Tsukuba University, Doctorial Program of Mathematics, Associate professor, 数学系, 助教授 (00243006)
TANIGUCHI Masahiko  Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50108974)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1999: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1998: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsSchrodinger equation / Dirac equation / Maxwell equation / Elliptic system / Unique continuation / Propagation of singularities / Smoothing effects / Microlocal analysis / 楕円形方程式系 / 一意持続性 / 解の一意接続性 / シュレーディンガー方程式 / 解の特異性の伝播
Research Abstract

The head investigator Okaji has investigated two fundamental properties, strong unique continuation property and propagation of singularities, of solutions to partial differential equations. As for the first property, he has treated elliptic systems of first order equations which are important in mathematical physics like Dirac operator or time harmonic Maxwell equations. In a joint work with De Carli, he obtained a nice condition which assures the strong unique continuation property for the Dirac operator with singular potential of Coulomb type. Furthermore, he has shown that the time harmonic Maxwell equation in non-isotropic and non-uniform continuous media has the strong continuation property if its coefficients are continuously differentiable.
As for the second topic, he has invented a new approach to the study of propagation of singularities of solutions to Schrodinger equations. This approach is based on a microlocal conservation law satisfied by the Wigner transformation of the solution. It is strongly connected to the wave packet transform of the solutions. As applications, he can clarify how propagate microlocal singularities of solutions to Schrodinger equations with vector potential as well as electric potential which may grow at the infinity. It includes smoothing effects, reconstruction of singularities and creation of singularities from oscillatory initial data.

Report

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

    (27 results)

All Other

All Publications (27 results)

  • [Publications] Takashi Okaji: "Strong Unique Continuation Property for the Dirac Equation"Publ.RIMS Kyoto Univ.. 35. 825-846 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 大鍛冶 隆司: "対称構造と偏微分方程式の解の特異性の伝播"数理解析研究所講究録. 1056. 75-96 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Shin'ichi Doi: "Commutator algebra and abstract smoothing effect"Journal of Functional Analysis. 168. 428-469 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Masahiko Taniguchi: "A condition of quasiconformal extendability"Proc.Japan Acad.Ser.A Math.Sci.. 75. 58-60 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Masahiko Taniguchi: "On topological completeness of decorated exponential families"Sci.Bull.of Josai Univ.. 4. 1-10 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Masahiko Taniguchi: "Holomorphic Dynamics"Cambridge Univ.Press. 338 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 重川 一郎: "確率解析,岩波講座現代数学の展開"岩波書店. 192 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takashi Okaji: "Strong Unique Continuation Property for the Dirac Equation"Publ. RIMS Kyoto Univ.. 35-6. 825-846 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takashi Okaji: "Symmetric structure and propagation of singularities of solutions to partial differential equations"Surikaisekikenkyujyo-Kokyuroku. 1056. 75-96 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Shin'ichi Doi: "Commutator algebra and abstract smoothing effect"Journal of Functional Analysis. 168-2. 428-469 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Masahiko, Taniguchi: "A condition of quasiconformal extendability"Proc. Japan Acad. Ser. A Math. Sci.. 75. 58--60 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Masahiko, Taniguchi: "On topological completeness of decorated exponential families"Sci. Bull. of Josai Univ.. 4. 1--10 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Masahiko Taniguchi,: "Holomorohic dynamics"Cambrige Univ. Press. 338 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ichiro Shigekawa: "Stochastic analysis"Iwanami Shoten. 192 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takashi Okaji: "Strong unique continuation theorems for Schrounger operators from a sphere"Houston Journal of Mathematics. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] Takashi Okaji: "Strong Unique Continuation Property for the Dirac Equation"Publ.RIMS Kyoto Univ.. 35・6. 825-846 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 大鍛冶 隆司: "対称構造と偏微分方程式にの解の特異性の伝播"数理解析研究所講究録. 1056. 75-96 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Shin'ichi Doi: "Commutator algebra and abstract smoothing effect"Journal of Functional Analysis. 168・2. 428-469 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Masahiko Taniguchi: "A condition of quasiconformal extendibility"Proc.Japan Acad.Ser.A Math.Sci. 75. 58-60 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Masahiko Taniguchi: "On topological completeness of decorated exponential families"Sci.Bull.of Josai Univ.. 4. 1-10 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Masahiko Taniguchi: "Holomorphic Dynamics"Cambridge Univ.Press. 338 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 重川 一郎: "確立解析、岩波講座現代数学の展開"岩波書店. 192 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Taniguchi: "On topological Completeness of decorated exponential families" Sci.Bull.of Josai Univ.4(Special Issue). 1-10 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] P.Aye-T.Nishida: "Heat convection of compressible fluid" Recent Developments in Domain Decomposition Methods and Flow Problems. 11. 107-115 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 大鍛冶 隆司: "対称構造と偏微分方程式の解の特異性の伝播" 数理解析研究所講究録. 1056. 75-96 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Matsuzaki-M.Taniguchi: "Hyperbolic manifolds and Kleinian Groups" Oxford Univ.Press, 253 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 重川 一郎: "確率解析、岩波講座現代数学の展開" 岩波書店, 192 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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