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

• Project/Area Number: 15K04962
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Kumamoto 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: ¥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.

Research Products

• [Int'l Joint Research] Aalto University(Finland)
• [Int'l Joint Research]
• [Journal Article] Regularity for the evolution of p-harmonic maps (2018)
  • Author(s): Masashi Misawa
  • Journal Title: J. Differential Equations Volume: 264 Pages: 1716-1749
  • Peer Reviewed
• [Journal Article] Global existence for the heat flow of symphonic maps into spheres (2018)
  • Author(s): Masashi Misawa, Nobumitsu Nakauchi
  • Journal Title: Advance in Differential Equations Volume: accepted for publication
  • Peer Reviewed
• [Journal Article] Local regularity and compactness for the p-harmonic map heat flows (2017)
  • Author(s): Masashi Misawa
  • Journal Title: Adv. Calc. Var. Volume: 64
  • Peer Reviewed
• [Journal Article] A Holder continuity of symphonic maps into the spheres (2016)
  • 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
  • Peer Reviewed
• [Journal Article] Gradient H\"older regularity for nonlinear parabolic systems of p-Laplacian type (2016)
  • Author(s): Corina Karim, Masashi Misawa
  • Journal Title: Differential Integral Equations Volume: 29, no. 3-4 Pages: 201-228
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] H\"older regularity for singular parabolic systems of p-Lapalcian type (2015)
  • Author(s): Corina Karim, Masashi Misawa
  • Journal Title: Adv. Differential Equations Volume: 20, no. 7-8 Pages: 741-772
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Global existence and regularity for the p-harmonic flow (2018)
  • Author(s): Masashi Misawa
  • Organizer: Conference on 'Metrics and Measures', Tohoku Univeristy, 2018/1/8-12
  • Invited
• [Presentation] Regularity for a doubly nonlinear parabolic equation and its application (2018)
  • Author(s): Masashi Misawa
  • Organizer: Matuyama kaiseki seminar 2018, Ehime University
  • Invited
• [Presentation] A doubly nonlinear degenerate and singular parabolic equation and a geometric flow (2018)
  • Author(s): Masashi Misawa
  • Organizer: Workshop, Critical exponent and nonlinear evolution equations 2018, Tokyo University of Science
  • Invited
• [Presentation] Monotonicity estimate and global existence for the p-harmonic flow (2018)
  • Author(s): Masashi Misawa
  • Organizer: RIMS Conference, 2018/2/5-8, Kyoto
  • Invited
• [Presentation] Regularity theory for nonlinear degenerate and singular parabolic systems and uts application to geometric flows (2017)
  • 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
  • Invited
• [Presentation] Global existence and partial regularity for the p-harmonic flow (2017)
  • Author(s): Masashi Misawa
  • Organizer: Free Boundary Problems and Nonlinear PDEs, Hokkaido University, 2017/9/26-28
  • 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
  • Invited
• [Presentation] Monotonicity type estimate and regularity criterion for p-harmonic map heat flows (2016)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Local regularity theorem for the evolution of p-harmonic maps (2016)
  • Author(s): Masashi Misawa
  • Organizer: 「Nonlinear PDEs」-Aspect of Regularity and Asymptotocs(Korean-Japan Int. Workshops)
  • Place of Presentation: Toya Sun Palace
  • 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
  • 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
  • Invited
• [Presentation] Local regularity criterion for the p-harmonic map heat flows (2016)
  • Author(s): Masashi Misawa
  • Organizer: 「PDE Workshop at Sendai」
  • Place of Presentation: Kawai Hall, Tohoku University
  • Int'l Joint Research / Invited
• [Presentation] On a degenerate type variational problem and its heat flow (2015)
  • 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
  • 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
  • Invited
• [Presentation] On a degenerate elliptic system and its heat flow (2015)
  • 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
  • Int'l Joint Research / Invited
• [Presentation] The evolution of p-harmonic maps (2015)
  • 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
  • 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, Kyoto (2017)
• [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