Analysis on the behavior and the singularity of solutions to fully nonlinear elliptic and parabolic partial differential equations

• Project/Area Number: 25400169
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Hiroshima University
• Principal Investigator: Kazuhiro Takimoto 広島大学, 理学研究科, 准教授 (00363044)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• 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

Outline of Final Research Achievements

We study the solvability of the boundary value problem, and the behavior and the singularity 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. Also, we obtain Bernstein type theorems for the generalized parabolic k-Hessian equation.

Research Products

• [Journal Article] Entire solutions to generalized parabolic k-Hessian equations (2016)
  • Author(s): Saori Nakamori and Kazuhiro Takimoto
  • Journal Title: Geometric Properties for Parabolic and Elliptic PDE's, Springer Proceedings in Mathematics & Statistics Volume: 176 Pages: 173-190
  • DOI: 10.1007/978-3-319-41538-3_11
  • ISBN: 9783319415369, 9783319415383
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A Bernstein type theorem for parabolic k-Hessian equations (2015)
  • Author(s): Saori Nakamori and Kazuhiro Takimoto
  • Journal Title: Nonlinear Analysis: Theory, Methods & Applications Volume: 117 Pages: 211-220
  • DOI: 10.1016/j.na.2015.01.010
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Entire solutions to generalized parabolic k-Hessian equations (2016)
  • Author(s): Kazuhiro Takimoto
  • Organizer: 11th AIMS International Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Hyatt Regency Orlando （アメリカ）
  • Year and Date: 2016-07-01
  • Int'l Joint Research / Invited
• [Presentation] 放物型k-Hessian方程式に対するBernstein型定理 (2016)
  • Author(s): 滝本和広
  • Organizer: 偏微分方程式の最大値原理とその周辺
  • Place of Presentation: 北海道大学
  • Year and Date: 2016-03-04
  • Invited
• [Presentation] Bernstein type theorem for some types of parabolic Hessian equations (2016)
  • Author(s): Kazuhiro Takimoto
  • Organizer: 第33回九州における偏微分方程式研究集会
  • Place of Presentation: 九州大学西新プラザ
  • Year and Date: 2016-01-27
  • Int'l Joint Research / Invited
• [Presentation] On a Bernstein type theorem for some types of parabolic Hessian equations (2015)
  • Author(s): Kazuhiro Takimoto
  • Organizer: RIMS研究集会「偏微分方程式の漸近問題と粘性解」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-12-04
  • Int'l Joint Research / Invited
• [Presentation] Entire solutions to the generalized parabolic k-Hessian equation (2015)
  • Author(s): 滝本和広
  • Organizer: 熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2015-11-07
  • Invited
• [Presentation] Bernstein type theorem for the generalized parabolic k-Hessian equation (2015)
  • Author(s): 滝本和広
  • Organizer: 京都大学NLPDEセミナー
  • Place of Presentation: 京都大学
  • Year and Date: 2015-10-23
  • Invited
• [Presentation] Entire solutions to some types of parabolic Hessian equations (2015)
  • Author(s): 滝本和広
  • Organizer: 日本数学会2015年度秋季総合分科会（函数方程式論分科会）
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-15
  • Invited
• [Presentation] Entire solution to the generalized parabolic k-Hessian equation (2015)
  • Author(s): Kazuhiro Takimoto
  • Organizer: Geometric Properties for Parabolic and Elliptic PDE's, 4th Italian-Japanese Workshop
  • Place of Presentation: Grand Hotel San Pietro （イタリア）
  • Year and Date: 2015-05-27
  • Int'l Joint Research / Invited
• [Presentation] Bernstein type theorem for the parabolic k-Hessian equation (2015)
  • Author(s): 中森さおり，滝本和広
  • Organizer: 日本数学会 2015 年度年会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-22
• [Presentation] Entire solution to the parabolic k-Hessian equation (2015)
  • Author(s): 滝本和広
  • Organizer: 南大阪応用数学セミナー
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2015-01-31
• [Presentation] Bernstein type theorem for some parabolic Hessian equation (2014)
  • Author(s): Kazuhiro Takimoto
  • 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: 2014-11-15
• [Presentation] Entire solution to the parabolic k-Hessian equation (2014)
  • Author(s): 滝本和広
  • Organizer: 岐阜数理科学セミナー
  • Place of Presentation: 岐阜大学
  • Year and Date: 2014-10-24
• [Presentation] Bernstein type theorem for some fully nonlinear PDEs (2014)
  • Author(s): Kazuhiro Takimoto
  • Organizer: 10th AIMS International Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Universidad Autonoma de Madrid（スペイン）
  • Year and Date: 2014-07-07
• [Presentation] The dynamics of a parabolic quasilinear boundary value problem associated to the mean curvature operator
  • Author(s): Kazuhiro Takimoto
  • Organizer: 3rd Italian-Japanese Workshop on Geometric Properties for Parabolic and Elliptic PDE's
  • Place of Presentation: 東京工業大学
• [Presentation] On the dynamics of a quasilinear parabolic problem related to the mean curvature operator
  • Author(s): 滝本和広
  • Organizer: 三大学偏微分方程式セミナー
  • Place of Presentation: 日本大学