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Research on structures of solutions for geometric variational problems

Research Project

Project/Area Number 15540214
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) OTSUKI Nobukazu  T.U.S., Faculty of Science and Technology, Professor, 理工学部, 教授 (80112895)
KUBAYASHI Tako  T.U.S., Faculty of Science and Technology, Professor, 理工学部, 教授 (90178319)
NAGASAWA Takeyuki  Saitama University, Faculty of Science, Professor, 理学部, 教授 (70202223)
OGASAWA Masao  T.U.S., Faculty of Science and Technology, Assistant, 理工学部, 助手 (50408704)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
KeywordsVariational problems / Partial regularity / VMO / Harmonic maps / 変分間題 / 正則性
Research Abstract

The aim of this research project is to investigate structures of solutions for geometric variational problems. While we were studying harmonic maps into Finsler manifolds, the necessity of the research on the variational functionals with singularities occurred. So, in this research project, we considered, as important points, regularity of the solutions for variational problems with singularities or weak solutions of partial differential equations with singular coefficients. For this purpose, we investigated partial regularity of minimizers for functionals with VMO (Vanishing Mean Oscillation)-coefficients in cooperation with Prof.Maria Alessandra Ragusa (Universita di Catania (Italy)). As results of cooperation with M.A.Ragusa, we got some results on partial regularity of minimizers. Namely, we proved that if u is a minimizer of certain functional with VMO-coefficients then u satisfies "u is Holder continuous except a subset of the domain whose m-2-ε dimensional Hausdorff measure is 0, where m is the dimension of the domain".
On the other hand, each researchers investigated their own problems : T.Nagasawa studied Helfrich variational problem which is one of mathematical models for shape transformation theory of human red blood cells. The existence of associated gradient flow was proved locally for arbitrary initial data, and globally near spheres. M.Ogawa studied free boundary problems for flows of an incompressible ideal fluid. He showed the unique existence of the solution, locally in time, even if the initial surface and the bottom are uneven.

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (15 results)

All 2006 2005 2004 Other

All Journal Article (14 results) Publications (1 results)

  • [Journal Article] On the existence of solutions of the Helfrich flow and itscenter manifold near spheres2006

    • Author(s)
      Yoshihito Kohsaka, Takeyuki Nagasawa
    • Journal Title

      Differential Integral Equations 19

      Pages: 121-142

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the existence of solutions of the Helfrich flow and Its center manifold near spheres2006

    • Author(s)
      Yoshihito Kohsaka, Takeyuki Nagasawa
    • Journal Title

      Differential Integral Equations 19

      Pages: 121-142

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the existence of solutions of the Helfrich flow and its center manifold near spheres2006

    • Author(s)
      Y.Kohsaka, T.Nagasawa
    • Journal Title

      Differential Integral Equations 19(2)

      Pages: 121-142

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Partial regularity of the minimizers of quadratic functionals with VMO coefficients2005

    • Author(s)
      Maria Alessandra Ragusa, Atsushi Tachikawa
    • Journal Title

      J. Lond. Math. Soc., II. Ser. 72

      Pages: 609-620

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On continuity of minimizers for certain quadratic functionals2005

    • Author(s)
      Maria Alessandra Ragusa, Atsushi Tachikawa
    • Journal Title

      J. Math. Soc. Japan 57

      Pages: 691-700

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Partial regularity of the minimizers of quadratic functionals with VMO coefficients2005

    • Author(s)
      Maria Alessandra Ragusa, Atsushi Tachikawa
    • Journal Title

      J.Lond.Math.Soc., II.Ser. 72

      Pages: 609-620

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On continuity of minimizers for certain quadratic Functionals2005

    • Author(s)
      Maria Alessandra Ragusa, Atsushi Tachikawa
    • Journal Title

      J.Math.Soc.Japan 57

      Pages: 691-700

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Partial regularity of the miniraizers of quadratic functionals with VMO coefficients2005

    • Author(s)
      M.A.Ragusa, A.Tachikawa
    • Journal Title

      Journal of the London Mathematical Society 72

      Pages: 609-620

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On continuity of minimizers for certain quadratic growth functionals2005

    • Author(s)
      M.A.Ragusa, A.Tachikawa
    • Journal Title

      Journal of the Mathematical Society of Japan 57

      Pages: 691-700

    • NAID

      10017177729

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Interior estimates in Campanato spaces related to quadratic functionals2004

    • Author(s)
      M.A.Ragusa, A.Tachikawa
    • Journal Title

      数理解析研究所講究録 1405

      Pages: 54-65

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Free surface motion of an incompressible ideal fluid

    • Author(s)
      Masao Ogawa
    • Journal Title

      Mathematishe Annalenに掲載予定

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Periodic vortical flows of an incompressible ideal fluid with free boundary

    • Author(s)
      Masao Ogawa, Atusi Tani
    • Journal Title

      Proceedings of the Conference on Hyperbolic Problems 2004に掲載予定

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Free surface motion of an incompressible ideal fluid

    • Author(s)
      Masao Ogawa
    • Journal Title

      Mathematische Annalen (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Periodic vortical flows of an incompressible ideal fluid with free boundary

    • Author(s)
      Masao Ogawa, Atusi Tani
    • Journal Title

      Proceedings of the Conference on Hyperbolic Problems 2004 (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] Atsushi Tachikawa: "A partial Holder regularity result for harmonic maps into a Finsler manifold"数理解析研究所講究録. 1347. 167-178 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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