Analysis on Fully Nonlinear Elliptic and Parabolic Partial Differential Equations and its Applications

• Project/Area Number: 22740091
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Hiroshima University
• Principal Investigator: TAKIMOTO Kazuhiro 広島大学, 大学院・理学研究科, 准教授 (00363044)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 完全非線形偏微分方程式 / 境界値問題 / 粘性解 / 解の存在と一意性 / 関数方程式 / 解析学 / 完全非線形偏微方程式

Research Abstract

We study the solvability of the boundary value problem and the behavior of solutions for fully nonlinear elliptic and parabolic partial differential equations. We also aim for the investigation of nonlinear phenomena. New results are obtained on the uniqueness of solutions to the so-called k-curvature equation, which is a fully nonlinear partial differential equation having geometric structure, and on the dynamics of the parabolic quasilinear problem of mean curvature type.

Research Products

• [Journal Article] Uniqueness of boundary blowup solutions to k-curvature equation (2013)
  • Author(s): Saori Nakamori and Kazuhiro Takimoto
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 399 Pages: 496-504
  • Peer Reviewed
• [Journal Article] A weighted quasilinear equation related to the mean curvature operator (2012)
  • Author(s): Santiago Cano-Casanova, Julian Lopez-Gomez and Kazuhiro Takimoto
  • Journal Title: Nonlinear Analysis Volume: 75 Pages: 5905-5923
  • Peer Reviewed
• [Journal Article] A quasilinear parabolic perturbation of the linear heat equation (2012)
  • Author(s): Santiago Cano-Casanova, Julian Lopez-Gomez and Kazuhiro Takimoto
  • Journal Title: Journal of Differential Equations Volume: 252 Pages: 323-343
  • Peer Reviewed
• [Journal Article] A quasilinear parabolic perturbation of the linear heat equation (2012)
  • Author(s): Santiago Cano-Casanova, Julian Lopez-Gromez, Kazuhiro Takimoto
  • Journal Title: Journal of Diffrential Equations Volume: 252 Pages: 323-343
  • Peer Reviewed
• [Journal Article] Removability of level sets for two classes of fully nonlinear equations (2010)
  • Author(s): Kazuhiro Takimoto
  • Journal Title: 京都大学数理解析研究所講究録「現象解析と関数方程式の新展開 (New Developments of Functional Equations in Mathematical Analysis)」 Volume: 1702 Pages: 158-171
• [Presentation] The dynamics of a quasilinear equation related to the mean curvature operator (2012)
  • Author(s): 滝本和広
  • Organizer: 東北大学非線形偏微分方程式ワークショップ
  • Place of Presentation: 東北大学
  • Year and Date: 2012-09-27
• [Presentation] Bernstein type theorem for (k,l)-Hessian quotient equation (2012)
  • Author(s): 中森さおり, 滝本和広
  • Organizer: 日本数学会 2012 年度年会(函数方程式論分科会)
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-27
• [Presentation] Bernstein type theorem for (k,l)-Hessian quotient equation (2012)
  • Author(s): 中森さおり, 滝本和広
  • Organizer: 日本数学会2012年度年会
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-27
• [Presentation] The dynamics of a quasilinear equation associated to the mean curvature operator (2012)
  • Author(s): Kazuhiro Takimoto
  • Organizer: 武漢大学数学与統計学院学術報告 (Symposia)
  • Place of Presentation: 武漢大学(中華人民共和国)
  • Year and Date: 2012-03-09
• [Presentation] The existence and uniqueness of boundary blowup solutions to k-curvature equation (2012)
  • Author(s): Kazuhiro Takimoto
  • Organizer: 東工大数理解析研究会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2012-02-06
• [Presentation] Bernstein type theorem for (k,l)-Hessian quotient equation (2012)
  • Author(s): 中森さおり, 滝本和広
  • Organizer: 平成 23 年度日本数学会中国・四国支部例会
  • Place of Presentation: 岡山大学
  • Year and Date: 2012-01-22
• [Presentation] Bernstein type theorem for (k,l)-Hessian quotient equation (2012)
  • Author(s): 中森さおり, 滝本和広
  • Organizer: 平成23年度日本数学会中国・四国支部例会
  • Place of Presentation: 岡山大学
  • Year and Date: 2012-01-22
• [Presentation] The dynamics of a parabolic quasilinear equation associated to the mean curvature operator (2012)
  • Author(s): 滝本 和広
  • Organizer: 東北大学非線形偏微分方程式ワークショップ
  • Place of Presentation: 東北大学
  • Invited
• [Presentation] Global solutions to some types of Hessian quotient equations (2011)
  • Author(s): 滝本和広
  • Organizer: 広島応用解析セミナー(第16回)
  • Place of Presentation: 広島大学
  • Year and Date: 2011-08-31
• [Presentation] Bernstein type theorem for Hessian quotient equations, INdAM Workshop (Second Italian-Japanese Workshop) (2011)
  • Author(s): Kazuhiro Takimoto
  • Organizer: Geometric Properties for Parabolic and Elliptic PDE's
  • Place of Presentation: Palazzone di Cortona(イタリア)
  • Year and Date: 2011-06-22
• [Presentation] Bernstein type theorem for Hessian quotient equations (2011)
  • Author(s): Kazuhiro Takimoto
  • Organizer: INdAM Workshop (Second Italian-Japanese Workshop), Geometric Properties for Parabolic and Elliptic PDE's
  • Place of Presentation: Palazzone di Cortona(イタリア)
  • Year and Date: 2011-06-22
• [Presentation] The existence and uniqueness of boundary blowup solutions to k-curvature equation (2011)
  • Author(s): 滝本和広
  • Organizer: 解析ゼミ・デラックス!
  • Place of Presentation: 埼玉大学
  • Year and Date: 2011-03-09
• [Presentation] On the existence and uniqueness of boundary blowup solutions to k-curvature equation (2010)
  • Author(s): Kazuhiro Takimoto
  • Organizer: Sino-Chilean Conference on Nonlinear Partial Differential Equations and Nonlinear Analysis
  • Place of Presentation: 武漢大学(中華人民共和国)
  • Year and Date: 2010-12-09
• [Presentation] On the existence and uniqueness of boundary blowup solutions to k-curvature equation (2010)
  • Author(s): 滝本和広
  • Organizer: 洞爺解析セミナー
  • Place of Presentation: 洞爺山水ホテル和風
  • Year and Date: 2010-09-29
• [Presentation] Uniqueness of a boundary blowup solution to k-curvature equation (2010)
  • Author(s): 滝本和広
  • Organizer: 日本数学会 2010 年度秋季総合分科会(函数方程式論分科会)
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] Uniqueness of a boundary blowup solution to k-curvature equation (2010)
  • Author(s): 滝本和広
  • Organizer: 日本数学会2010年度秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] On the uniqueness of a solution to the boundary blowup problem for curvature equation (2010)
  • Author(s): Kazuhiro Takimoto
  • Organizer: 8thAIMS International Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Dresden University of Technology(ドイツ)
  • Year and Date: 2010-05-26
• [Presentation] On the uniqueness of a solution to the boundary blowup problem for curvature equation (2010)
  • Author(s): K.Takimoto
  • Organizer: 8^<th> AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Dresden University of Technology(ドイツ)
  • Year and Date: 2010-05-26