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Monotonicity estimates and local regularity and singularity for the p-harmonic flows

Research Project

Project/Area Number 15K04962
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionKumamoto University

Principal Investigator

Masashi Misawa  熊本大学, 大学院先端科学研究部(理), 教授 (40242672)

Co-Investigator(Renkei-kenkyūsha) YAMAURA Yoshihiko  日本大学, 文理学部, 教授 (90255597)
Project Period (FY) 2015-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords偏微分方程式 / 退化特異放物型方程式系 / 正則性特異性 / 調和写像 / 調和写像熱流 / p調和写像 / p調和写像熱流 / p調和方程式 / 退化特異楕円型 / p調和写像流 / 退化特異放物型
Outline of Final Research Achievements

The p-harmonic flow is the heat flow for the p-harmonic maps,the critical
points for p-energy for maps beteween two smooth compact Riemannian manifolds, and given by the degenerate and singular parabolic system of 2nd ordered partial differential equations of so-called p-Laplcian type. We study the regularity and singularity of solutions of the p-harmonic flows. Our main result is the local regularity theorem, which holds true uniformly for the p-energy bounded, regular solutions of the p-harmonic flows. Based on this local regularity theorem, we prove that the weak limits of p-energy bounded, regular p-harmonic flows are partial regular weak solutions of the p-harmonic flows, with the almost optimal size estimation by Hausdorff measure, of their singular sets, the exceptional sets of regularity.

Report

(4 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (26 results)

All 2018 2017 2016 2015 Other

All Int'l Joint Research (2 results) Journal Article (6 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 6 results) Presentation (16 results) (of which Int'l Joint Research: 6 results,  Invited: 16 results) Funded Workshop (2 results)

  • [Int'l Joint Research] Aalto University(Finland)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research]

    • Related Report
      2016 Research-status Report
  • [Journal Article] Regularity for the evolution of p-harmonic maps2018

    • Author(s)
      Masashi Misawa
    • Journal Title

      J. Differential Equations

      Volume: 264 Pages: 1716-1749

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global existence for the heat flow of symphonic maps into spheres2018

    • Author(s)
      Masashi Misawa, Nobumitsu Nakauchi
    • Journal Title

      Advance in Differential Equations

      Volume: accepted for publication

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Local regularity and compactness for the p-harmonic map heat flows2017

    • Author(s)
      Masashi Misawa
    • Journal Title

      Adv. Calc. Var.

      Volume: 64

    • Related Report
      2017 Annual Research Report 2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] A Holder continuity of symphonic maps into the spheres2016

    • Author(s)
      Masashi Misawa and Nobumitsu Nakauchi
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 55 Issue: 1

    • DOI

      10.1007/s00526-015-0940-0

    • Related Report
      2016 Research-status Report 2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Gradient H\"older regularity for nonlinear parabolic systems of p-Laplacian type2016

    • Author(s)
      Corina Karim, Masashi Misawa
    • Journal Title

      Differential Integral Equations

      Volume: 29, no. 3-4 Pages: 201-228

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] H\"older regularity for singular parabolic systems of p-Lapalcian type2015

    • Author(s)
      Corina Karim, Masashi Misawa
    • Journal Title

      Adv. Differential Equations

      Volume: 20, no. 7-8 Pages: 741-772

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Global existence and regularity for the p-harmonic flow2018

    • Author(s)
      Masashi Misawa
    • Organizer
      Conference on 'Metrics and Measures', Tohoku Univeristy, 2018/1/8-12
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Regularity for a doubly nonlinear parabolic equation and its application2018

    • Author(s)
      Masashi Misawa
    • Organizer
      Matuyama kaiseki seminar 2018, Ehime University
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] A doubly nonlinear degenerate and singular parabolic equation and a geometric flow2018

    • Author(s)
      Masashi Misawa
    • Organizer
      Workshop, Critical exponent and nonlinear evolution equations 2018, Tokyo University of Science
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Monotonicity estimate and global existence for the p-harmonic flow2018

    • Author(s)
      Masashi Misawa
    • Organizer
      RIMS Conference, 2018/2/5-8, Kyoto
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Regularity theory for nonlinear degenerate and singular parabolic systems and uts application to geometric flows2017

    • Author(s)
      Masashi Misawa
    • Organizer
      Functional Analysis and Partial Differential Equations, 2017 CIMPA-MONGOLIA research school, Cetre International de Mathematiques Pures et Appliquees, Khovd University, Mongolia, 2017/7/17-28
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Global existence and partial regularity for the p-harmonic flow2017

    • Author(s)
      Masashi Misawa
    • Organizer
      Free Boundary Problems and Nonlinear PDEs, Hokkaido University, 2017/9/26-28
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] 時間発展p調和作用素に対する局所正則性とその応用2017

    • Author(s)
      三沢 正史
    • Organizer
      「関数空間の構造とその周辺」(RIMS研究集会)
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Monotonicity type estimate and regularity criterion for p-harmonic map heat flows2016

    • Author(s)
      Masashi Misawa
    • Organizer
      Variational Problems and Nonlinear Partial Differential Equations 2016
    • Place of Presentation
      東京理科大学理工学部数学教室(千葉県野田市)
    • Year and Date
      2016-03-24
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Local regularity theorem for the evolution of p-harmonic maps2016

    • Author(s)
      Masashi Misawa
    • Organizer
      「Nonlinear PDEs」-Aspect of Regularity and Asymptotocs(Korean-Japan Int. Workshops)
    • Place of Presentation
      Toya Sun Palace
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] p調和写像熱流の単調性評価と局所正則性定理2016

    • Author(s)
      三沢 正史
    • Organizer
      研究集会「第12回 非線形の諸問題」
    • Place of Presentation
      かんぽの宿 湯田
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] p調和写像熱流に対する単調型評価と正則性2016

    • Author(s)
      三沢 正史
    • Organizer
      「微分方程式セミナー」
    • Place of Presentation
      大阪大学数学教室
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Local regularity criterion for the p-harmonic map heat flows2016

    • Author(s)
      Masashi Misawa
    • Organizer
      「PDE Workshop at Sendai」
    • Place of Presentation
      Kawai Hall, Tohoku University
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On a degenerate type variational problem and its heat flow2015

    • Author(s)
      Masashi Misawa
    • Organizer
      2015 NIMS Hot Topics Workshop on Regularity Theory on Elliptic and Parabolic Equations
    • Place of Presentation
      Sangsan Mathematical Science Bulding, Seoul National University, Seoul, Korea
    • Year and Date
      2015-12-10
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] ある幾何学に現れる退化型変分問題とその熱流について2015

    • Author(s)
      三沢 正史
    • Organizer
      南大阪応用数学セミナー
    • Place of Presentation
      大阪市立大学理学部数学教室(大阪市住吉区)
    • Year and Date
      2015-11-28
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] On a degenerate elliptic system and its heat flow2015

    • Author(s)
      Masashi Misawa
    • Organizer
      Nonlinear PDE Workshop at Tohoku University
    • Place of Presentation
      Kawai Hall, Faculty of Science, Tohoku University, Sendai, Japan
    • Year and Date
      2015-09-25
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The evolution of p-harmonic maps2015

    • Author(s)
      Masashi Misawa
    • Organizer
      Geometric Properties for Parabolic and Elliptic PDE's, 4th Italian-Japanese Workshop
    • Place of Presentation
      Palinuro, Italy
    • Year and Date
      2015-05-27
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Regularity, singularity and long time behavior for partial differential equations with conservation laws, RIMS conference, 2017/6/7-9, Kyoto2017

    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] 「Regularity, sungularity and long time behavior for paertial differential equations with conservation laws」2016

    • Place of Presentation
      Research Institute for Mathematical Sciences, Kyoto University
    • Year and Date
      2016-06-06
    • Related Report
      2016 Research-status Report

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Published: 2015-04-16   Modified: 2019-03-29  

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