Fundamental theory for viscosity solutions of fully nonlinear equations and its applications

• Project/Area Number: 23340028
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tohoku University
• Principal Investigator: Koike Shigeaki 東北大学, 理学研究科, 教授 (90205295)
• Co-Investigator(Kenkyū-buntansha): 小川 卓克 東北大学, 理学(系)研究科(研究院), 教授 (20224107) ; 石井 克幸 神戸大学, 海事科学研究科(研究院), 教授 (40232227) ; 石井 仁司 早稲田大学, 教育・総合科学学術院, 教授 (70102887) ; 長井 英生 関西大学, 工学部, 教授 (70110848) ; 三上 敏夫 津田塾大学, 学芸学部, 教授 (70229657) ; 石毛 和弘 東北大学, 理学(系)研究科(研究院), 教授 (90272020) ; 岡部 真也 東北大学, 理学(系)研究科(研究院), 准教授 (70435973)
• Project Period (FY): 2011-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥18,200,000 (Direct Cost: ¥14,000,000、Indirect Cost: ¥4,200,000); Fiscal Year 2015: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2014: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2013: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2012: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2011: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
• Keywords: 粘性解 / 最大値原理 / 比較原理 / ハルナック不等式 / 完全非線形 / 障害問題 / 非局所作用素 / 平均曲率流 / ABP最大値原理 / 弱Harnack不等式 / 自由境界問題 / 処罰法 / ヘルダー連続 / 分数冪ラプラス作用素 / 冪乗ラプラス作用素 / 弱ハルナック不等式 / 完全非線形方程式 / 動的計画原理 / 最適制御理論 / レヴィ過程 / 局所最大値原理

Outline of Final Research Achievements

We obtained comparison principle for unbounded viscosity solutions of degenerate elliptic PDE with superlinear gradient terms. We presented a representation formula for viscosity solutions of integro-differential equations of Isaacs type. We established the local maximum principle fro Lp-viscosity solutions of fully nonlinear uniformly elliptic PDE with unbounded coefficients to the first derivatives. We discussed regularity and large time behavior of viscosity solutions of integro-differential equations with coercive first derivative terms. We obtained existence and uniqueness of entire solutions of fully nonlinear elliptic equations with superlinear growth in the first derivatives. We showed the Lipschitz continuity of viscosity solutions of mean curvature flow equations with bilateral obstacles.

Research Products

• [Journal Article] Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term (2016)
  • Author(s): G. Galise, S. Koike, O. Ley, A. Vitolo
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 441 Issue: 1 Pages: 194-210
  • DOI: 10.1016/j.jmaa.2016.03.083
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Remarks on the comparison principle for quasilinear PDE with no zeroth order terms (2015)
  • Author(s): Shigeaki Koike, Takahiro Kosugi
  • Journal Title: Commun. Pure Appl. Anal. Volume: 14 Issue: 1 Pages: 133-1423
  • DOI: 10.3934/cpaa.2015.14.133
  • Peer Reviewed / Open Access
• [Journal Article] Regularity results and large time behavior for integro-differential equations with coercive Hamiltonians (2015)
  • Author(s): G. Barles, S. Koike, O. Ley and E. Topp
  • Journal Title: Calculus of Variations, and Partial Differential Equations Volume: 掲載確定
  • Peer Reviewed / Open Access
• [Journal Article] On the ABP maximum principle for Lp-viscosity solutions of fully nonlinear PDE (2014)
  • Author(s): S. Koike
  • Journal Title: Nonlinear Dynamics in Partial Differential Equations,Mathematics, Advanced Studies in Pure Mathematics Volume: 64 Pages: 113-124
  • Peer Reviewed
• [Journal Article] Representation formulas for solutions of Isaacs integro-PDE (2013)
  • Author(s): S. Koike, A. Swiech
  • Journal Title: Indiana University Mathematical Journal Volume: 62 Issue: 5 Pages: 1473-1502
  • DOI: 10.1512/iumj.2013.62.5109
  • Peer Reviewed
• [Journal Article] On the ABP maximum principle and applications (2013)
  • Author(s): S. Koike
  • Journal Title: 数理解析研究所講究録 Volume: 1845 Pages: 107-120
• [Journal Article] Representation formulas for solutions of Isaacs integro-PDE (2013)
  • Author(s): S. Koike and A. Swiech
  • Journal Title: Indiana University Mathematical Journal Volume: 未定
  • Peer Reviewed
• [Journal Article] Lcal maximum principle for Lp viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients (2012)
  • Author(s): S. Koike, A. Swiech
  • Journal Title: Communications in Pure and Applied Analysis Volume: 11, No5 Issue: 5 Pages: 1897-1910
  • DOI: 10.3934/cpaa.2012.11.1897
  • Peer Reviewed
• [Journal Article] Comparison principle for unbounded viscosity solutions of degenerate elliptic PDEs with gradient superlinear terms (2011)
  • Author(s): S. Koike
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 381 Issue: 1 Pages: 110-120
  • DOI: 10.1016/j.jmaa.2011.03.009
  • Peer Reviewed
• [Presentation] 粘性解のABP最大値原理とその応用 (2017)
  • Author(s): 小池茂昭
  • Organizer: 日本数学会2017年度年会
  • Place of Presentation: 首都大学東京
  • Year and Date: 2017-03-24
  • Invited
• [Presentation] Fully nonlinear uniformly elliptic/parabolic PDE with unbounded ingredients (2016)
  • Author(s): Shigeaki Koike
  • Organizer: Hamilton-Jacobi Equations New trends and applications
  • Place of Presentation: Rennes大学
  • Year and Date: 2016-05-30
  • Int'l Joint Research / Invited
• [Presentation] ABP maximum principle for fully nonlinear elliptic PDE with unbounded ingredients (2014)
  • Author(s): S. Koike
  • Organizer: Colloquium
  • Place of Presentation: 復旦大学、中国
  • Year and Date: 2014-12-08
  • Invited
• [Presentation] 微分積分方程式の解のヘルダー連続性 (2014)
  • Author(s): 小池茂昭
  • Organizer: 北九州地区における偏微分方程式研究集会
  • Place of Presentation: 小倉
  • Year and Date: 2014-11-29
  • Invited
• [Presentation] A note on comparison principle for quasilinear equations (2014)
  • Author(s): S. Koike
  • Organizer: AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Madrid, Spain
  • Year and Date: 2014-07-08
  • Invited
• [Presentation] On the ABP maximum principle and applications (2011)
  • Author(s): 小池 茂昭
  • Organizer: 幾何学的偏微分方程式における保存則と正則性の研究
  • Place of Presentation: 数理解析研究所
  • Invited
• [Presentation] On the ABP maximum principle for Lp-viscosity solutions of fully nonlinear PDE (2011)
  • Author(s): 小池 茂昭
  • Organizer: The 4th MSJ-SI Nonlinear Dynamics and PDE国際研究集会
  • Place of Presentation: 九州大学
  • Invited
• [Presentation] On the ABP maximum principle for viscosity solutions (2011)
  • Author(s): 小池 茂昭
  • Organizer: 広島微分方程式研究会
  • Place of Presentation: 広島大学
  • Invited
• [Presentation] Lp粘性解のABP最大値原理とその応用 (2011)
  • Author(s): 小池 茂昭
  • Organizer: 応用解析研究会
  • Place of Presentation: 早稲田大学
  • Invited
• [Book] 粘性解 -比較原理を中心に- (2016)
  • Author(s): 小池茂昭
  • Total Pages: 216
  • Publisher: 共立出版
• [Book] リメディアル数学 (2011)
  • Author(s): 泉屋周一、上江洌達也、小池茂昭、重本和泰、徳永浩雄
  • Publisher: 数学書房
• [Remarks] http://www.math.tohoku.ac.jp/~koike/kenkyu/