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A versatile study of positive solutions to parabolic and elliptic partial differential equations

Research Project

Project/Area Number 11440039
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionTokyo Institute of Technology

Principal Investigator

MURATA Minoru  Graduate School of Science and Engineering, Tokyo Institute of Technology Professor, 大学院・理工学研究科, 教授 (50087079)

Co-Investigator(Kenkyū-buntansha) KURATA Kazuhiro  Tokyo Metropolitan University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (10186489)
AIKAWA Hiroaki  Shimane University, Interdisciplinary Faculty of Science and Engineering, Professor, 総合理工学部, 教授 (20137889)
ISHIGE Kazuhiro  Nagoya University, Graduate School of Mathematics, Tokyo Institute of Technology Associate Professor, 大学院・多元数理科学研究科, 助教授 (90272020)
SHIGA Hiroshige  Graduate School of Science and Engineering, Tokyo Institute of Technology Professor, 大学院・理工学研究科, 教授 (10154189)
UCHIYAMA Kouhei  Graduate School of Science and Engineering, Tokyo Institute of Technology Professor, 大学院・理工学研究科, 教授 (00117566)
志賀 徳造  東京工業大学, 大学院・理工学研究科, 教授 (60025418)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2000: ¥2,500,000 (Direct Cost: ¥2,500,000)
Keywordsparabolic PDE / elliptic PDE / positive solution / Cauchy problem / uniqueness / fundamental solution / perturbation theory / Martin boundary / 楕円形偏微分方程式 / 放物型方程式 / 楕円型方程式 / 歪積 / ポテンシャル論
Research Abstract

M.Murata and K.Ishige studied uniqueness of nonnegative solutions of the Cauchy problem to second order parabolic equations on Riemannian manifolds and domains of R^n, and gave a sharp and general sufficient condition for the uniqueness via asymptotic properties at infinity of the equation and manifolds (to appear in Ann. Scuola Normale Sup. Pisa.). This is a simple and general result which unifies all previous results on the uniqueness.
M.Murata investigated the structure of positive solutions of second order elliptic equations in skew product form, and determined the Martin boundary and Martin kernel by exploiting and developping perturbation theory and estimates of fundamental solutions for parabolic equations. This result was reviewed by such experts as Y.Pinchover, A.Grigor'yan, V.Maz'ya, S.Gardiner, and has been submitted.
K.Ishige studied uniqueness of nonnegative solutions of the Dirichlet and Neumann boundary value problems to second order parabolic equations on domains of R^n, and gave sharp sufficient conditions for the uniqueness via asymptotic properties at infinity of the equation and domains. For the Dirichlet problem, he established an optimal uniqueness theorem (Journal of Differential Equations, 158(1999), 251 -290). As for the Neumann problem, he revealed a large difference from the Dirichlet problem and established a uniqueness theorem for domains belonging to a class wider than that for the Dirichlet probelm.

Report

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

    (34 results)

All Other

All Publications (34 results)

  • [Publications] K.Ishige and M.Murata: "Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains"Ann.Scuola Normale Sup.Pisa.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Ishige: "An intrinsic metric approach to uniqueness of the positive Dirichlet problem for parabolic equations in cylinders"Journal of Differential Equations. 158. 251-290 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Aikawa: "Integrability of superharmonic functions in a John domain"Proc.Amer.Math.Soc.. 128. 195-201 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Aikawa: "Boundary Harnack principle and Martin boundary for a uniform domain"J.Math.Soc.Japan. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Kurata: "On Doubling Property for Non-Negative Weak solutions of Elliptic and Parabolic PDE"Israel J.Math.. 115. 285-302 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Kurata: "Existence and semi-classical limit of the least energy solution to a nonlinear Schrodinger equation with electromagnetic fields"Nonlinear Analysis. 41. 763-778 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kohei Uchiyama: "Uniqueness of solutions to the initial value problem for an integro-differential equation"Diff.Int.Eqs.. 13. 401-422 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kohei Uchiyama: "A non-linear evolution equation driven by a logarithmic potential"Bull.London Math.Soc.. 32. 353-363 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Shiga and H.Tanigawa: "Projective structures on Riemann surfaces with discontinuous holonomies"Trans.Amer.Math.Soc.. 352(2). 813-823 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Shiga: "On holomorphic families of rational maps : Finiteness, Rigidity and Stability"Kodai Math.J.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 相川弘明(編集): "ポテンシャル論とその関連分野"京都大学数理解析研究所. 181 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Ishige and M.Murata: "Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains"to appear in Ann. Scuola Normale Sup. Pisa..

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Ishige: "An intrinsic metric approach to uniqueness of the positive Dirichlet problem for parabolic equations in cylinders"Journal of Differential Equations. 158. 251-290 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Aikawa: "Integrability of superharmonic functions in a John domain"Proc. Amer. Math. Soc.. 128. 195-201 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Aikawa: "Boundary Harnack principle and Martin boundary for a uniform domain"J.Math. Soc. Japan. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Kurata: "On Doubling Property for Non-Negative Weak solutions of Elliptic and Parabolic PDE"Israel J.Math.. 115. 285-302 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Kurata: "Existence and semi-classical limit of the least energy solution to a nonlinear Schrodinger equation with electromagnetic fields"Nonlinear Analysis. 41. 763-778 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kohei Uchiyama: "Uniqueness of solutions to the initial value problem for an integro-differential equation"Diff. Int. Eqs.. 13. 401-422 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kohei Uchiyama: "A non-linear evolution equation driven by a logarithmic potential"Bull. London Math. Soc.. 32. 353-363 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Shiga and H.Tanigawa: "Projective structures on Riemann surfaces with discontinuous holonomies"Trans. Amer. Math. Soc.. 351 (2). 813-823 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Shiga: "On holomorphic families of rational maps : Finiteness, Rigidity and Stability."to appear in Kodai. Math. J..

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Ishige & M.Murata: "Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains"Ann.Scuola Normale Sup.Pisa. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Ishige: "An intrinsic metric approach to uniqueness of the positive Dirichlet problem for parabolic equations in cylinders"J.Diff.Eq.. 158. 251-290 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Aikawa: "Integrability of superharmonic functions in a John domain"Proc.Amer.Math.Soc.. 128. 195-201 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Aikawa: "Boundary Harnack principle and Martin boundary for a uniform domain"J.Math.Soc.Japan. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Kurata: "On doubling property for non-negative weak solutions of elliptic and parabolic PDE"Israel J.Math.. 115. 285-302 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Uchiyama: "A non-linear evolution equation driven by a logarithmic potential"Bull.London Math.Soc.. 32. 353-363 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 相川弘明(編集): "ポテンシャル論とその関連分野"京都大学数理解析研究所. 181 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Uchiyama: "Uniqueness of solutions to the initial value problem for an integro-differential equation (to appear)"Diff.Int.Eqs. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Shiga and H.Tanigawa: "Projective structures on Riemann surfaces with discontinuous holonomies"Trans.Amer.Math.Soc.. 351. 813-823 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Z.li,T.Shiga and L.Yao: "A Reversibility Problem for Fleming-Viot Processes"Electronic Communication in Probability. 4. 71-82 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Aikawa: "Integrability of superharmonic functions in a John domain"Proc.Amer.Math.Soc.. 128. 195-201 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Aikawa: "Boundary Harnack principle and Martin boundary for a uniform domain (to appear)"J.Math.Soc.Japan.

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Kurata: "An estimate on the heat kernel of magnetic schodinger operators and uniformlly elliptic operators with non-negative potentials (to appear)"J.London Math.Soc.

    • Related Report
      1999 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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