Analysis of solutions to partial differential equations with piecewise smooth coefficients, and its application to inverse problems

• Project/Area Number: 23740110
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: University of Hyogo
• Principal Investigator: NAGAYASU Sei 兵庫県立大学, 物質理学研究科, 講師 (90455684)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 偏微分方程式 / 逆問題 / 安定性評価

Research Abstract

Now we consider partial differential equations (PDEs) with some parameter. The parameter gives some influences for the solutions to PDEs. One of the aim of this research is to analyse such influences. In this study period, I analysed stability estimates for inverse problems of PDEs with frequency. In particular, we treated the wave equation and the Schroedinger equation. In consequence, I obtained some stability estimates. These stability estimates suggest us that stability becomes better when influence becomes large. However, unfortunately right now we do not know this estimate is optimal or not. Hence I would like to continue to analyse this problem.

Research Products

• [Journal Article] Increasing stability of the inverse boundary value problem for the Schrodinger equation (2014)
  • Author(s): Victor Isakov, Sei Nagayasu, Gunther Uhlmann, Jenn-Nan Wang
  • Journal Title: Contemporary Mathematics Volume: 615 (印刷中)
  • Peer Reviewed
• [Journal Article] Increasing stability of the inverse boundary value problem for the Schroedinger equation (2014)
  • Author(s): Victor Isakov, Sei Nagayasu, Gunther Uhlmann, Jenn-Nan Wang
  • Journal Title: Contemporary Mathematics Volume: 615
  • Peer Reviewed
• [Journal Article] A gradient estimate for solutions to parabolic equations with discontinuous coefficients (2013)
  • Author(s): Jishan Fan, Kyoungsun Kim, Sei Nagayasu, Gen Nakamura
  • Journal Title: Electronic Journal of Differential Equations 2013 Volume: No.93 Pages: 1-24
  • URL: http://ejde.math.txstate.edu/Volumes/2013/93/abstr.html
  • Peer Reviewed
• [Journal Article] Increasing stability in an inverse problem for the acoustic equation (2013)
  • Author(s): Sei Nagayasu, Gunther Uhlmann, Jenn-Nan Wang
  • Journal Title: Inverse Problems Volume: 29
  • URL: http://stacks.iop.org/IP/29/025012
  • Peer Reviewed
• [Journal Article] A gradient estimate for solutions to parabolic equations with discontinuous coefficients (2013)
  • Author(s): Jishan Fan, Kyoungsun Kim, Sei Nagayasu, Gen Nakamura
  • Journal Title: Electronic Journal of Differential Equations Volume: 2013
  • Peer Reviewed
• [Presentation] Increasing stability of the inverse boundary value problem for the Schrodinger equation (2013)
  • Author(s): 永安聖 (Victor Isakov 氏, Gunther Uhlmann 氏, Jenn-Nan Wang 氏との共同研究)
  • Organizer: 微分方程式セミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2013-05-31
• [Presentation] Increasing stability in inverse problems for some equations (2013)
  • Author(s): 永安聖 (Victor Isakov 氏, Gunther Uhlmann 氏, Jenn-Nan Wang 氏との共同研究)
  • Organizer: 南大阪応用数学セミナー
  • Place of Presentation: 大阪府立大学
  • Year and Date: 2013-05-18
• [Presentation] Increasing stability in an inverse problem for some equations (2013)
  • Author(s): 永安聖 (Victor Isakov 氏, Gunther Uhlmann 氏, Jenn-Nan Wang 氏との共同研究)
  • Organizer: 逆問題とその周辺分野に関するミニワークショップ
  • Place of Presentation: 東京大学
  • Year and Date: 2013-03-27
• [Presentation] Increasing stability in an inverse problem for the acoustic equation and the Schrodinger equation (2012)
  • Author(s): Sei Nagayasu (Victor Isakov 氏, Gunther Uhlmann 氏, Jenn-Nan Wang 氏との共同研究)
  • Organizer: Taiwan-Japan Joint Conference on PDE and Analysis 2012
  • Place of Presentation: 国立台湾大学 (National Taiwan University, 台湾)
  • Year and Date: 2012-12-27
• [Presentation] Increasing stability in an inverse problem for the acoustic equation (2012)
  • Author(s): Sei Nagayasu (Gunther Uhlmann 氏, Jenn-Nan Wang 氏との共同研究)
  • Organizer: International Conference on Inverse Problems and Related Topics 2012
  • Place of Presentation: 東南大学(Southeast University, 中国)
  • Year and Date: 2012-10-25
• [Presentation] 区分的に滑らかな係数を持つ偏微分方程式の解の勾配評価と偏微分方程式の逆問題における介在物再構成の問題 (2012)
  • Author(s): 永安聖 (講演内容にはGunther Uhlmann 氏, Jenn-Nan Wang 氏との共同研究, 及びJishan Fan 氏, Kyoungsun Kim 氏, 中村玄氏との共同研究を含む)
  • Organizer: 逆問題への応用を意図した解析学の研究(RIMS 共同研究)
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] 区分的に滑らかな係数を持つ偏微分方程式の解の勾配評価と偏微分方程式の逆問題における介在物再構成の問題 (2012)
  • Author(s): 永安聖
  • Organizer: 逆問題への応用を意図した解析学の研究
  • Place of Presentation: 京都大学数理解析研究所 (京都府)
  • Invited
• [Presentation] Increasing stability in an inverse problem for the acoustic equation (2011)
  • Author(s): 永安聖 (Gunther Uhlmann 氏, Jenn-Nan Wang 氏との共同研究)
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 東京大学
  • Year and Date: 2011-12-17
• [Presentation] Increasing stability in an inverse problem for the acoustic equation (2011)
  • Author(s): 永安聖 (Gunther Uhlmann 氏, Jenn-Nan Wang 氏との共同研究)
  • Organizer: 微分方程式セミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2011-11-11
• [Presentation] Increasing stability in an inverse problem for the acoustic equation (2011)
  • Author(s): 永安聖
  • Organizer: 微分方程式の総合的研究(招待講演)
  • Place of Presentation: 東京大学(東京都)
• [Presentation] Increasing stability in inverse problems for some equations
  • Author(s): 永安聖
  • Organizer: 南大阪応用数学セミナー
  • Place of Presentation: 大阪府立大学(大阪府)
  • Invited
• [Presentation] Increasing stability of the inverse boundary value problem for the Schroedinger equation
  • Author(s): 永安聖
  • Organizer: 微分方程式セミナー
  • Place of Presentation: 大阪大学(大阪府)
  • Invited
• [Presentation] Increasing stability in an inverse problem for the acoustic equation
  • Author(s): Sei Nagayasu
  • Organizer: International Conference on Inverse Problems and Related Topics 2012
  • Place of Presentation: 東南大学 (南京，中国)
• [Presentation] Increasing stability in an inverse problem for the acoustic equation and the Schroedinger equation
  • Author(s): Sei Nagayasu
  • Organizer: Taiwan-Japan Joint Conference on PDE and Analysis 2012
  • Place of Presentation: 国立台湾大学 (台北，台湾)
• [Presentation] Increasing stability in an inverse problem for some equations
  • Author(s): Sei Nagayasu
  • Organizer: 逆問題とその周辺分野に関するミニワークショップ
  • Place of Presentation: 東京大学 (東京都)
  • Invited