Regularity for the evolutionary p-Laplace operator and global existence of the p-harmonic map flows

• Project/Area Number: 24540215
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Global analysis
• Research Institution: Kumamoto University
• Principal Investigator: MISAWA MASASHI 熊本大学, 自然科学研究科, 教授 (40242672)
• Co-Investigator(Renkei-kenkyūsha): YAMAURA Yoshihiko 日本大学, 文理学部, 教授 (90255597)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 偏微分方程式 / 退化特異放物型作用素 / 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調和写像流 / 変分問題 / 正則性

Outline of Final Research Achievements

We have been studied the regularity problem for p-harmonic map heat flows. The p-harmonic maps are critical points for p-energy for maps beteween two smooth compact Riemannian manifolds and so, the solution of the Euler Lagrange equation of p-energy, which is 2nd ordered degenerate and singular elliptic partial differential equations of so-called p-Laplcian type. In this research project we will study the existence and regularity of solutions of the p-harmonic map equations. In particular, we study the evolution equation, called the p-harmonic map heat flow. Our main result is to show the regularity of small solutions of the p-harmonic map heat flows. Under this main result, we also to show the regularity and existence of a global small solution of the m-harmonc map type heat flows with space dimension m. The asymptotic behavior to the stationary solution is also shown.

Research Products

• [Journal Article] H\"older regularity for singular parabolic systems of p-Laplacian type (2015)
  • Author(s): Corina Karim, Masashi Misawa
  • Journal Title: Advances in Differential Equations Volume: to appear
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] The regularity for nonlinear parabolic systems of p-Laplacian type with critical growth (2014)
  • Author(s): C. Leone, M. Misawa, A. Verde
  • Journal Title: J. Differntial Equations Volume: 256 Pages: 2807-2845
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] The regularity for nonlinear parabolic systems of p-Laplacian type with critical growth (2014)
  • Author(s): C. Leone, M. Misawa, A, Verde
  • Journal Title: J. Differential Equations Volume: 256 Pages: 2807-2845
  • Peer Reviewed
• [Journal Article] L^2 boundedness for the 2D exterior problems for the semilinear heat and dissipative wave equations (2013)
  • Author(s): T. Kobayashi, M. Misawa
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B42 Pages: 1-11
  • Peer Reviewed
• [Journal Article] A Holder estimate for nonlinear parabolic systems of p-Laplacian type (2013)
  • Author(s): Masashi Misawa
  • Journal Title: J. Differential Equati ons Volume: 254 Issue: 2 Pages: 847-878
  • DOI: 10.1016/j.jde.2012.10.001
  • Peer Reviewed
• [Journal Article] Holder continuity of minimizing symphonic maps (2012)
  • Author(s): M. Misawa, N. Nakauchi
  • Journal Title: Nonlinear Anal. Volume: 75 Issue: 15 Pages: 5971-5974
  • DOI: 10.1016/j.na.2012.06.010
  • Peer Reviewed
• [Journal Article] A global existence result for the heat flow of higher dimensional H-systems (2011)
  • Author(s): C.Leone, M.Misawa, A, Verde
  • Journal Title: J.Math.Pures Appl. Volume: 97 Issue: 3 Pages: 282-294
  • DOI: 10.1016/j.matpur.2011.11.001
  • Peer Reviewed
• [Presentation] Monotonicity type estimate and regularity of the p-harmonic map heat flows (2014)
  • Author(s): Masashi Misawa
  • Organizer: 「台湾数学会, 関数方程式分科会」
  • Place of Presentation: 台湾成功大学
  • Year and Date: 2014-12-06 – 2014-12-07
  • Invited
• [Presentation] Regularity for the p-harmonic map heat flows (2014)
  • Author(s): Masashi Misawa
  • Organizer: 「The 9th East Asia Partial Differential Equations Conference」（Satelite of Seoul-ICM, Korea)
  • Place of Presentation: ホテル日航奈良
  • Year and Date: 2014-07-28 – 2014-07-31
  • Invited
• [Presentation] p調和作用素の正則性と単調性評価 (2014)
  • Author(s): 三沢 正史
  • Organizer: 「秋田における偏微分方程式論集中ワークショップ」
  • Place of Presentation: 秋田大学教育文化学部
  • Year and Date: 2014-07-11 – 2014-07-13
  • Invited
• [Presentation] p調和作用素と関連する幾何学的変分問題とその正則性問題 (2014)
  • Author(s): 三沢 正史
  • Organizer: 「函館における偏微分方程式論集中ワークショップ」
  • Place of Presentation: 公立はこだて未来大学システム情報学部
  • Year and Date: 2014-04-05 – 2014-04-06
  • Invited
• [Presentation] p調和写像熱流の正則性について (2013)
  • Author(s): 三沢 正史
  • Organizer: 「微分方程式の総合的研究」
  • Place of Presentation: 東京大学数理科学研究科
  • Invited
• [Presentation] Regularity problems for the evolution of p-harmonic maps (2013)
  • Author(s): Masashi Misawa
  • Organizer: Quasilinear PDEs & Game Theory
  • Place of Presentation: Uppsala University, Uppsala, Sweden
  • Invited
• [Presentation] On regularity conditions of the p-harmonic map flows (2013)
  • Author(s): Masashi Misawa
  • Organizer: Internatinal Seminar at 「Evolutonary problems」
  • Place of Presentation: Institute Mittag-Leffler, Sweden
  • Invited
• [Presentation] pラプラス作用素に対する存在, 正則性と関連する変分問題-エネルギー臨界型p調和方程式系の弱解の正則性とその応用 (2012)
  • Author(s): 三沢正史
  • Organizer: 第34回 発展方程式 若手セミナー特別講演「発展方程式若手セミナー」
  • Place of Presentation: タナベ湘南研修センター
  • Invited
• [Presentation] A regularity for nonlinear parabolic systems of p-Laplacian type with critical growth (2012)
  • Author(s): Masashi Misawa
  • Organizer: 「アールト大学数学解析セミナー」
  • Place of Presentation: アールト大学, Helsinki, Finland
  • Invited