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Behavior of spatial critical points and zeros of solutions of partial differential equations

Research Project

Project/Area Number 12440042
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionEhime University

Principal Investigator

SAKAGUCHI Shigeru  Ehime University, Faculty of Science, Professor, 理学部, 教授 (50215620)

Co-Investigator(Kenkyū-buntansha) IKEHATA Masaru  Gunma University, Faculty of Engineering, Professor, 工学部, 教授 (90202910)
HASHIMOTO Takahiro  Ehime University, Faculty of Science, Instructors, 理学部, 助手 (60291499)
YANAGI Shigenori  Ehime University, Faculty of Science, Associate Professor, 理学部, 助教授 (10253296)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥7,600,000 (Direct Cost: ¥7,600,000)
Fiscal Year 2002: ¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2001: ¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2000: ¥2,600,000 (Direct Cost: ¥2,600,000)
Keywordsheat equation / hot spot / isothermic surface / symmetry / initial-Dirichlet problem / sphere / hypersurface / polygon / 初期斉次Diricllet問題 / 非有界領域 / 空間臨界点 / 初期斉次デリクレ問題 / 双曲空間
Research Abstract

1. Let Ω be a domain in the N-dimensional Euclidean space, and consider the initial-Dirichlet problem for initial data being a positive constant. Suppose that D is a domain satisfying the interior cone condition and D^^-⊂Ω. We considered the question how the boundary ∂D is a stationary isothermic surface of the solution, and obtained the following two theorems : (i) Let Ω be either a bounded domain or an exterior domain satisfying the exterior sphere condition. If ∂D is a stationary isothermic surface, then ∂Ω must be a sphere. (ii) Let Ω be an unbounded domain satisfying the uniform exterior sphere condition, and suppose that ∂Ω contains a nonempty open subset where the principal curvatures of ∂Ω with respect to the exterior normal direction to ∂Ω are nonnegative. Furthermore, assume that, for any r > 0, ∂Ω contains the graph over a (N -1)-dimensional ball with radius r > 0. If ∂D is a stationary isothermic surface, then ∂Ω must be either a hyperplane or two parallel hyperplanes.
2. Th … More ere is a conjecture of Chamberland and Siegel (1997) concerning the hot spots of solutions of the heat equation. Let Ω be a bounded domain in the Euclidean space containing the origin, and consider the initial-Dirichlet problem for initial data being a positive constant. The conjecture stated that if the origin is a stationary hot spot, then Ω is invariant under the action of an essential subgroup G of orthogonal transformations. Concerning this conjecture, we obtained the following four theorems when the space dimension is two : (i) Let Ω be a triangle. If the origin is a stationary hot spot, then Ω must be an equilateral triangle centered at the origin. (ii) Let Ω be a convex quadrangle, then Ω must be a parallelogram centered at the origin. (iii) If the origin is a stationary hot spot, then Ω is not a non-convex quadrangle. (iv) Let Ω be a convex m-polygon ( m = 5 or 6 ). Suppose that the inscribed circle centered at the origin touches every side of Ω, and suppose that the origin is a stationary hot spot. Then, if m = 5, Ω must be a regular pentagon centered at the origin, and if m = 6, Ω must be invariant under the rotation of one of three angles, π/3, 2π/3, and π. Less

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (18 results)

All Other

All Publications (18 results)

  • [Publications] S.Sakaguchi: "Stationary critical points of the heat flow in spaces of constant curvature"Journal London Mathematical Society. 63. 400-412 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] R.Magnanini, S.Sakaguchi: "Stationary critical points of the heat flow in the plane"Journal d'Analyse Mathematique. 88. 383-396 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] R.Magnanini, S.Sakaguchi: "Matzoh ball soup: Heat conductors with a stationary isothermic surface"Annals of Mathematics. 156. 931-946 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] R.Magnanini, S.Sakaguchi: "On heat conductors with a stationary hot spot"Annali di Matematica pura ed applicata. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 坂口 茂: "拡散方程式の解の空間臨界点と零点の挙動"数学. 54. 249-264 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Sakaguchi: "Behavior of spatial critical points and zeros of solutions of diffusion equations"Sugaku Expositions. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Sakaguchi: "Stationary critical points of the heat flow in spaces of constant curvature"Journal London, Mathematical Society. 63-2. 400-412 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] R.Magnanini and S.Sakaguchi: "Stationary critical points of the heat flow in the plane"Journal d'Analyse Mathematique. 88. 383-396 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] R.Magnanini and S.Sakaguchi: "Matzoh ball soup : Heat conductors with a stationary isothermic surface"Annals of Mathematics. 156-3. 931-946 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] R.Magnanini and S.Sakaguchi: "On heat conductors with a stationary hot spot"Annali di Matematica pura ed applicata. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Sakaguchi: "Behavior of spatial critical points and zeros of solutions of diffusion equations"Sugaku (Japanese). 54-3. 249-264 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Sakaguchi: "Behavior of spatial critical points and zeros of solutions of diffusion equations"Sugaku Expositions (English translation). (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] R.Magnanini, S.Sakaguchi: "Matzoh ball soup : Heat conductors with a stationary isothermic surface"Annals of Mathematics. 156-3. 931-946 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] R.Magnanini, S.Sakaguchi: "On heat conductors with a stationary hot spot"Annali di Matematica pura ed applicata. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] 坂口 茂: "拡散方程式の解の空間臨界点と零点の挙動"数学. 54・3. 249-264 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Sakaguchi: "Behavior of spatial critical points and zeros of solutions of diffusion equations"Sugaku Expositions. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] R.Magnanini, S.Sakaguchi: "Stationary critical points of the heat flow in the plane"Journal d'Analyse Mathematique. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] Shigeru Sakaguchi: "Stationary critical points of the heat flow in spaces of constant curvature"Journal London Mathematical Society. (発表予定).

    • Related Report
      2000 Annual Research Report

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

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