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Research on the Regularity of Solutions for Geometric Variational Problems

Research Project

Project/Area Number 12640221
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTokyo University of Science

Principal Investigator

TACHIKAWA Atsushi  Tokyo University of Science(T.U.S.), Faculty of Science and Technology, Professor, 理工学部, 教授 (50188257)

Co-Investigator(Kenkyū-buntansha) FURUTANI Kenro  T.U.S., Fac. of Sci. and Tech., Professor, 理工学部, 教授 (70112901)
KOBAYASHI Takao  T.U.S., Fac. of Sci. and Tech., Professor, 理工学部, 教授 (90178319)
NAGASAWA Takeyuki  Saitarna Univ., Fac. of Science, Professor, 理学部, 教授 (70202223)
TANAKA Makiko  T.U.S., Fac. of Sci. and Tech., Professor, 理工学部, 助教授 (20255623)
YAMAZAKI Taeko  T.U.S., Fac. of Sci. and Tech., Associate Professor, 理工学部, 助教授 (60220315)
牛島 健夫  東京理科大学, 理工学部, 講師 (30339113)
小林 嶺道  東京理科大学, 理工学部, 教授 (70120186)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2000: ¥800,000 (Direct Cost: ¥800,000)
KeywordsVariattional Problems / regularity / Harmonic maps / Finsler manifold / energy functional
Research Abstract

The variational problems for the energy functional defined for maps between Riemmannian manifolds have been studied by many mathematicians.Namely, about harmonic maps between Riemannian manifolds, we have many deep results.Recently, some generalisations of harmonic maps attract the interest of several reseachers.In this research, we considered some generalised notion of harmonic maps and got some results on their regularity
In the year 2000, we treated harmonic maps with potentials, and got their existence and regularity results.In the year 2001, we considered more generalised notion of harmonic maps so-called F-farmonic tnays and get their existence and partial regularity results
In the year 2002, we treated harmonic maps into Finsler manidolds.Finsler manifolds are natural generalization of Riemannian ones.P.Centore defined the energy of a map between Finsler manifolds.We used Centores definition and specialized it for the case that the source manifold is Eucldean space. We calculated the Euler-Lagrange equation of it and got the equation for hannonic maps from an. Euclidean space R^m to a Finsler manifold. Moreover, we got a partial regularity result for energy minimizing map from R^m to a Finsler manifold for the case that m=3,4.More precisely, we proved that for the above cases the enegy minimizing maps are Wilder continuous outside the singular set whose Hausdorff dimension is less that m-2.

Report

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

    (26 results)

All Other

All Publications (26 results)

  • [Publications] Takeyuki Nagasawa, Atsushi Tachikawa: "Blow-up solutions for ordinary differential equations associated to harmonic maps and its applications"J.Math.Soc.Japan. 53. 485-500 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Atsushi Tachikawa: "Existence and regularity results for harmonic maps with potential."Tokyo J.Math.. 24. 195-204 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Atsushi Tachikawa: "Existence and regularity results for some variational problems related to harmonic maps."Nonlinear Anal.. 47. 1703-1714 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Atsushi Tachikawa: "A partial regularity result for harmonic maps into Finsler manifolds."Calc.Var.. 16. 217-224 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Nagasawa, K.Nakane, S.Omata: "Numerical computations for motion of vortices governed by a hyperbolic Ginzburg-Landau system"Nonlinear Anal.. 51・1. 67-77 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] P.Nagawawa, I.Takagi: "Bifurcating critical points of bending energy with constraints related to the shape of red blood cell"Cal.Var.Partial Differential Equations. 16・1. 63-111 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takeyuki NAGASAWA, Atsushi TACHIKAWA: "Blow-up solutions for ordinary differential equations associated to hannonic maps and its applications"J.Math.Soc.Japan. 53. 485-500 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Atsushi TACHIKAWA: "Existence and regularity results for harmonic maps with potential."Tokyo J.Math.. 24. 195-204 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Atsushi TACHIKAWA: "Existence and regularity results for some variational problems related to harmonic maps."Nonlinear Anal.. 47. 1703-1714 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Atsushi TACHIKAWA: "A partial regularity result for harmonic maps into Finsler manifolds."Calc.Var.(Erratum, Calc.Var.). 16,16. 217,225-224,226 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takeyuki NAGASAWA, Kazuaki NAKANE, Seiro OMATA: "Numerical computations for motion of vortices governed by a Hyperbolic Ginzburg-Landau system"Nonlinear Anal.. 51. 67-77 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Theyuki NAGASAWA, Izumi TAKAGI: "Bifurcating critical points of bending energy with constraints related to the shape of red blood cell,"Calc.Var.. 16. 63-111 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Nagasawa, K.Nakane, S.Omata: "Numerical computations for motion of vortices governed by a hyperbolic Ginzburg-Landau system"Nonlinear Anal.. 51・1. 67-77 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Nagawawa, I.Takagi: "Bifurcating critical points of bending energy with constraints related to the shape of red blood cell"Cal. Var. Partial Differential Equations. 16・1. 63-111 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Taeko Yamazaki: "Sufficient conditions on the forcing terms for the global solvability of dissipative Kirchhoff equations"Nonlinear Anal.. 51. 969-982 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Makiko Sumi Tanaka: "Subspaces in the category of symmetric spaces"Contemporary Mathematics. 308. 305-313 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Atsushi TACHIKAWA: "Existence and regularity results for harmonic maps with potential"Tokyo J.Math.. 24・1. 195-204 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Atsushi TACHIKAWA: "Existence and regularity results for some variational problems related to harmonic maps"Nonlinear Analysis. 47・3. 1703-1714 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Nagasawa, A.Tachikawa: "Blow-up selection for ordinary differential equations associated to harmonic maps and their applications"J.Math.Soc.JAPAN. 53・2. 485-500 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Takeyuki Nagasawa: "A new energy inequality and partial regularity for weak solations of Navier-Stokes equations"J.Math.Fluid Mech. 3・1. 40-56 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Takeyuki Nagasawa: "A refinement of the energy inequality for the Navier-Stokes equations"Nonlinear Analysis. 47・6. 4245-4256 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Atsushi TACHIKAWA: "Existence and regularity results for harmonic maps with potential"Tokyo J.Math.. 未定(to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Nagasawa-A.Tachikawa: "Blow-up solutions for ordinary differential equations associated to harmonic maps and its applications"J.Math.Soc.JAPAN. 未定(to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Furutani-N.Otsuki-B.Booss-Bavnbek: "Criss-Cross Reduction of the Maslov index and a proof of the Yoshida-Nicolaescu theorem"Tokyo.J.Math. 24・1(to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Nagasawa-I.Takagi: "Closed surfaces minimizing the bending energy under prescribed area and volume"International Conference on Differential Equations Berlin 1999. 1. 561-563 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Nagano.M.S.Tanaka (田中真紀子): "The involutions of Compact symmetric spaces,V"Tokyo J.Math.. 23. 403-416 (2000)

    • Related Report
      2000 Annual Research Report

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

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