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Research on the regularity of solutions for nonlinear partial differential equations related to variational problems

Research Project

Project/Area Number 22540207
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TACHIKAWA Atsushi  東京理科大学, 理工学部, 教授 (50188257)

Co-Investigator(Renkei-kenkyūsha) 長澤 壯之  埼玉大学, 理学研究科, 教授 (70202223)
Project Period (FY) 2010-04-01 – 2014-03-31
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords関数方程式 / 変分問題 / 弱解の正則性 / Finsler多様体 / p(x)-growth functional / 解析学 / non-standard grwoth / 非線形偏微分方程式 / 調和写像 / 偏微分方程式 / 正則性 / p(x)-growth
Research Abstract

The problems to find maps or functions that give critical points of a quantity under consideration are called variational problems. When we treat variational problems, we often employ the following 2-step procedure: first, we find a "solution" in the class of maps that are differentiable in certain generalized sense, and, as the second step, we prove that the "solution" is appropriately smooth. This second step is called "regularity problem", and in this research we treat "regularity problem". In general, when we consider "regularity problem", we often assume continuity or, more strongly, differentiability of coefficients. In this research, we tried to obtain some regularity results under weaker conditions on the smoothness of coefficients, and we have gotten some new results.

Report

(5 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • Research Products

    (21 results)

All 2014 2013 2012 2011 2010

All Journal Article (10 results) (of which Peer Reviewed: 10 results) Presentation (11 results) (of which Invited: 2 results)

  • [Journal Article] On the singular set of minimizers of p(x)−energies2014

    • Author(s)
      A. Tachikawa
    • Journal Title

      Carc. Val. Partial Differ. Equ.

      Volume: 50(1-2) Issue: 1-2 Pages: 145-169

    • DOI

      10.1007/s00526-013-0631-7

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] C^1,α-regularity of energy minimizing maps from a 2-dimentional domain into aフィンスラーspace2013

    • Author(s)
      A. Tachikawa
    • Journal Title

      Houston J. Math.

      Volume: 39(4) Pages: 1175-1186

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Partial regularity of p(x)-harmonic maps2013

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

      Trans. Amer. Math. Soc.

      Volume: 365(6) Issue: 6 Pages: 3329-3353

    • DOI

      10.1090/s0002-9947-2012-05780-1

    • Related Report
      2013 Annual Research Report 2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] On interior regularity of minimizers of p(x)-energy functionals2013

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

      Nonlinear Anal.

      Volume: 93 Pages: 162-167

    • DOI

      10.1016/j.na.2013.07.023

    • Related Report
      2013 Annual Research Report 2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] $C^{1,\alpha}$-regularity of energy minimizing maps from a 2-dimentional domain into a Finsler space2013

    • Author(s)
      Atsushi Tachikawa
    • Journal Title

      Houston Journal of Mathematics

      Volume: 39

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Partial regularity of p(x)-harmonic maps2013

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

      Transactions of America Mathematical Society

      Volume: 365

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] C1,α-Regularity of energy minimizing maps from a 2-dimensional domain into a Finsler space2013

    • Author(s)
      Atsushi Tachikawa (Atsushi Tachikawa)
    • Journal Title

      Houston Journal of Mathematics

      Volume: 未定

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence and regularity of weakly harmonic maps into a Finsler manifold with a special structure2012

    • Author(s)
      A. Tachikawa
    • Journal Title

      Bull. Lond. Math. Soc.

      Volume: 44(5) Issue: 5 Pages: 1020-1033

    • DOI

      10.1112/blms/bds030

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Estimates of the derivatives of minimizers of a special class of variational integrals2011

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

      Discrete Contin. Dyn. Syst.

      Volume: 31(4) Issue: 4 Pages: 1411-1425

    • DOI

      10.3934/dcds.2011.31.1411

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Estimates of the derivatives of minimizers of a special class of variational integrals2011

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

      Discrete Contin.Dyn.Syst.Ser.A

      Volume: 31 Pages: 1411-1425

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Presentation] p(x)-調和写像の正則性について2014

    • Author(s)
      立川篤
    • Organizer
      日本数学会2014年度年会関数方程式分科会特別講演
    • Place of Presentation
      学習院大学
    • Year and Date
      2014-03-17
    • Related Report
      2013 Final Research Report
  • [Presentation] p(x)-調和写像の正則性について2014

    • Author(s)
      立川 篤
    • Organizer
      日本数学会2014年度年会・関数方程式分科会・特別講演
    • Place of Presentation
      学習院大学
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Partial regularity of p(x)-harmonic maps2012

    • Author(s)
      立川篤
    • Organizer
      RIMS研究集会"幾何学的偏微分方程式に対する保存則と正則性特異性の研究"
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2012-06-13
    • Related Report
      2013 Final Research Report
  • [Presentation] p(x)-調和写像の正則性について2012

    • Author(s)
      立川篤
    • Organizer
      仙台小研究集会–西川青季教授定年退職記念–
    • Place of Presentation
      東北大学理学部数理科学記念館(川井ホール)
    • Year and Date
      2012-03-16
    • Related Report
      2013 Final Research Report
  • [Presentation] Partial regularity of p(x)-harmonic maps2012

    • Author(s)
      立川 篤 (立川 篤)
    • Organizer
      RIMS研究集会”幾何学的偏微分方程式に対する保存則と正則性特異性の研究"
    • Place of Presentation
      RIMS(京都大学数理解析研究所)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] p(x)-調和写像の部分正則性について2011

    • Author(s)
      立川篤
    • Organizer
      北海道大学偏微分方程式セミナー
    • Place of Presentation
      北海道大学理学部3号館202教室
    • Year and Date
      2011-11-07
    • Related Report
      2013 Final Research Report
  • [Presentation] On the regularity of harmonic maps from a 2 dimensionalリーマンmanifold into aフィンスラーmanifold2011

    • Author(s)
      立川篤
    • Organizer
      さいたま数理解析セミナー(第8回)
    • Place of Presentation
      埼玉大学大宮ソニックシティカレッジ
    • Year and Date
      2011-07-29
    • Related Report
      2013 Final Research Report
  • [Presentation] On the regularity of harmonic maps from a 2 dimensional Riemannian manifold into a Finsler manifold2011

    • Author(s)
      立川篤
    • Organizer
      さいたま数理解析セミナー
    • Place of Presentation
      埼玉大学大宮ソニックシティカレッジ
    • Year and Date
      2011-07-29
    • Related Report
      2011 Annual Research Report
  • [Presentation] フィンスラー多様体への調和写像の部分正則性について2011

    • Author(s)
      立川篤
    • Organizer
      神楽坂解析セミナー
    • Place of Presentation
      東京理科大学神楽坂校舎
    • Year and Date
      2011-04-23
    • Related Report
      2013 Final Research Report 2011 Annual Research Report
  • [Presentation] On the regularity of energy minimizing maps into Finsler manifolds2011

    • Author(s)
      立川篤
    • Organizer
      Workshop on Geometric Analysis, Sendai 2011
    • Place of Presentation
      東北大学理学部数学棟
    • Year and Date
      2011-02-23
    • Related Report
      2010 Annual Research Report
  • [Presentation] On the regularity of harmonic maps into Finsler manifolds2010

    • Author(s)
      立川篤
    • Organizer
      微分方程式の総合的研究
    • Place of Presentation
      京都大学大学院理学研究科
    • Year and Date
      2010-12-18
    • Related Report
      2010 Annual Research Report

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Published: 2010-08-23   Modified: 2019-07-29  

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