Studies on nonlinear partial differential equations via potential analysis

• Project/Area Number: 22740081
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Hiroshima University (2013); Akita University (2010-2012)
• Principal Investigator: HIRATA Kentaro 広島大学, 理学(系)研究科(研究院), 准教授 (30399795)
• Project Period (FY): 2010-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: ポテンシャル論 / 偏微分方程式 / 実解析 / 複素解析 / 熱方程式 / 半線形楕円型方程式 / 境界挙動 / 半線形放物型方程式 / 特異点 / 熱核 / 大域的評価 / ハルナック不等式 / 除去可能集合 / ハウスドルフ次元

Research Abstract

We have investigated, from the point of view of potential theory, some relationships among geometric properties of domains, nonlinearity of equations and properties of solutions of nonlinear partial differential equations. In nonsmooth domains, we obtained several results concerning a boundary growth estimate, a Harnack type inequality, the existence of nontangential limits, a characterization of removable sets for Holder continuous solutions of semilinear elliptic equations, a removability theorem for semilinear heat equations and a sharp global estimate of the heat kernel.

Research Products

• [Journal Article] Removable singularities of semilinear parabolic equations (2014)
  • Author(s): K. Hirata
  • Journal Title: Proceedings of the American Mathematical Society Volume: Vol.142 Issue: 1 Pages: 157-171
  • DOI: 10.1090/s0002-9939-2013-11739-9
  • Peer Reviewed
• [Journal Article] Two sided global estimates of heat kernels in Lipschitz domains (2013)
  • Author(s): K. Hirata
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: Vol.43 Pages: 29-45
  • URL: http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu-j.html
  • Peer Reviewed
• [Journal Article] Admissible boundary limits of Green potentials satisfying nonlinear inequalities in the unit ball of C^n (2013)
  • Author(s): K. Hirata
  • Journal Title: Topics in Finite or Infinite Dimensional Complex Analysis Pages: 111-121
  • Peer Reviewed
• [Journal Article] Two sided global estimates of heat kernels in Lipschitz domains (2013)
  • Author(s): Kentaro Hirata
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: 43 Pages: 29-45
  • Peer Reviewed
• [Journal Article] Admissible boundary limits of Green potentials satisfying nonlinear inequalities in the unit ball of C^n (2013)
  • Author(s): Kentaro Hirata
  • Journal Title: Topics in finite or infinite dimensional complex analysis Volume: なし Pages: 111-121
  • Peer Reviewed
• [Journal Article] Boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains, Trans (2011)
  • Author(s): K. Hirata
  • Journal Title: Amer. Math. Soc Volume: 363 Issue: 08 Pages: 4007-4025
  • DOI: 10.1090/s0002-9947-2011-05071-3
  • Peer Reviewed
• [Journal Article] Removable sets for continuous solutions of semilinear elliptic equations (2011)
  • Author(s): Kentaro Hirata
  • Journal Title: Manuscripta Math. Volume: 135 Issue: 1-2 Pages: 245-262
  • DOI: 10.1007/s00229-011-0440-2
  • Peer Reviewed
• [Journal Article] Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains (2010)
  • Author(s): K. Hirata
  • Journal Title: Journal of the Mathematical Society of Japan Volume: Vol.62 Pages: 1043-1068
  • NAID: 10027870995
  • URL: http://projecteuclid.org/euclid.jmsj/1288703096
  • Peer Reviewed
• [Journal Article] Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains (2010)
  • Author(s): Kentaro Hirata
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 62 Pages: 1043-1068
  • NAID: 10027870995
  • Peer Reviewed
• [Presentation] Removable singularities of solutions of semilinear heat equations (2013)
  • Author(s): 平田賢太郎
  • Organizer: 第100回熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2013-11-09
• [Presentation] Removable singularities of solutions of semilinear heat equations (2013)
  • Author(s): K. Hirata
  • Organizer: 3^<rd> Italian-Japanese Workshop on Geometric Properties for Parabolic and Elliptic PDE's
  • Place of Presentation: 東京工業大学
  • Year and Date: 2013-09-03
• [Presentation] Removable singularities of solutions of semilinear heat equations (2013)
  • Author(s): 平田賢太郎
  • Organizer: 2013年度ポテンシャル論研究集会
  • Place of Presentation: 北海道大学
  • Year and Date: 2013-08-29
• [Presentation] 熱方程式の除去可能な特異点に関するHsuの結果のポテンシャル論的証明 (2013)
  • Author(s): 平田賢太郎
  • Organizer: 北海道大学月曜解析セミナー
  • Place of Presentation: 北海道大学
  • Year and Date: 2013-07-22
• [Presentation] 非線形不等式を満たす優調和関数の境界挙動 (2013)
  • Author(s): 平田賢太郎
  • Organizer: 広島大学数学教室談話会
  • Place of Presentation: 広島大学
  • Year and Date: 2013-04-23
• [Presentation] 熱方程式の除去可能な特異点に関するHsuの結果のポテンシャル論的証明 (2012)
  • Author(s): 平田賢太郎
  • Organizer: 第7回なかもず解析セミナー
  • Place of Presentation: 大阪府立大学
  • Year and Date: 2012-11-30
• [Presentation] Heat kernel estimates and growth estimates of solutions of semilinear heat equations (2012)
  • Author(s): K. Hirata
  • Organizer: RIMS workshop : Potential Theory and its Related Fields
  • Place of Presentation: 京都大学
  • Year and Date: 2012-09-07
• [Presentation] Heat kernel estimates and growth estimates of solutions of semilinear heat equations (2012)
  • Author(s): Kentaro Hirata
  • Organizer: RIMS workshop ``Potential Theory and its Related Fields''
  • Place of Presentation: 京都大学
• [Presentation] 熱方程式の除去可能な特異点に関するHsuの結果のポテンシャル論的証明 (2012)
  • Author(s): 平田賢太郎
  • Organizer: なかもず解析セミナー
  • Place of Presentation: 大阪府立大学
  • Invited
• [Presentation] Admissible boundary limits of Green potentials satisfying nonlinear inequalities in the unit ball of C^n (2011)
  • Author(s): K. Hirata
  • Organizer: The 19th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications
  • Place of Presentation: 広島アステールプラザ
  • Year and Date: 2011-12-14
• [Presentation] Admissible boundary limits of Green potentials satisfying nonlinear inequalities in the unit ball of C^n (2011)
  • Author(s): Kentaro Hirata
  • Organizer: The 19th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications
  • Place of Presentation: Aster Plaza (広島)
  • Year and Date: 2011-12-14
• [Presentation] 非線形不等式を満たす優調和関数に対する非接極限定理 (2011)
  • Author(s): 平田賢太郎
  • Organizer: 2011年度ポテンシャル論研究集会
  • Place of Presentation: 岐阜大学
  • Year and Date: 2011-11-04
• [Presentation] 非線形不等式を満たす優調和関数に対する非接極限定理 (2011)
  • Author(s): 平田賢太郎
  • Organizer: ポテンシャル論研究集会
  • Place of Presentation: 岐阜大学
  • Year and Date: 2011-11-04
• [Presentation] Lipschitz領域において非線形不等式を満たす正値優調和関数に対するCarleson評価について (2011)
  • Author(s): 平田賢太郎
  • Organizer: 北海道大学月曜解析セミナー
  • Place of Presentation: 北海道大学
  • Year and Date: 2011-09-26
• [Presentation] 非線形不等式を満たす優調和関数の非接極限定理とCarleson評価 (2011)
  • Author(s): 平田賢太郎
  • Organizer: 第46回函数論サマーセミナー
  • Place of Presentation: かんぽの宿別府
  • Year and Date: 2011-09-13
• [Presentation] 半線形楕円型方程式の解の連続性と除去可能集合の関係 (2010)
  • Author(s): 平田賢太郎
  • Organizer: 2010年度ポテンシャル論研究集会
  • Place of Presentation: 大分大学
  • Year and Date: 2010-11-06
• [Presentation] Removable singularities of solutions of semilinear heat equations
  • Author(s): Kentaro Hirata
  • Organizer: 3rd Italian-Japanese Workshop on Geometric Properties for Parabolic and Elliptic PDE's
  • Place of Presentation: 東京工業大学
  • Invited
• [Remarks]
  • URL: http://edu.math.ie.akita-u.ac.jp/~hirata/