Unique continuation problems and complex phase methods in the theory of partial differential equations

• Project/Area Number: 22540185
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: TAKASHI OKAJI 京都大学, 理学(系)研究科(研究院), 准教授 (20160426)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); 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: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 解の一意接続性 / ディラック作用素 / 放物型方程式 / スペクトル問題 / 解析学 / 解析・評価 / 関数方程式論 / 数理物理学 / 函数解析 / スペクトル / コーシ-問題 / 解析接続 / 強一意接続性 / コーシー-グルサー問題 / Dirac作用素

Outline of Final Research Achievements

We investigated spectral properties of Dirac operators for the relativistic particles like electrons and obtained important results on its essential self-adjointness and the absence of eigenvalues at the thresholds. Moreover, we established the strong unique continuation property for parabolic operators of second order with singular potentials. In addition, we revealed a global structure of the singularities of solutions to the Cauchy problem for a partial differential equation in the whole complex domain.

Research Products

• [Journal Article] The Dirac Operator with Mass m0≧ 0 : Non-Existence of Zero Modes and of Threshold Eigenvalues (2015)
  • Author(s): Hubert Kalf, Takashi Okaji, and Osanobu Yamada
  • Journal Title: Documenta Mathematica Volume: 20 Pages: 37-64
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Le prolongement analytique de la solution du probleme de Cauchy pour un certain operateur differentiel a coefficients polynomiaux. II (2014)
  • Author(s): Yusaku Hamada, Takashi Okaji and Yoshitsugu Takei
  • Journal Title: Journal of Fixed Point Theory and Applications Volume: (online version) Issue: 1 Pages: 1-36
  • DOI: 10.1007/s11784-014-0152-9
  • Peer Reviewed
• [Journal Article] Le prolongement analytique de la solution du problème de Cauchy pour un certain opérateur différentiel a coefficients polynomiaux. II& III (2013)
  • Author(s): Y. Hamada, T. Okaji and Y. Takei
  • Journal Title: Journal of Fixed Point Theory and Applications Volume: 14巻 Issue: 1 Pages: 1-15
  • DOI: 10.1007/s11784-014-0153-8
  • Peer Reviewed
• [Journal Article] A note on unique continuation for parabolic operators (2013)
  • Author(s): Takashi Okaji
  • Journal Title: Studies in Phase Space Analysis with Applications to PDEs, Progr.Nonlinear Diff. Eq. Appl., Volume: 84
  • Peer Reviewed
• [Journal Article] Problème de Cauchy et Goursat analytique, Publications of RIMS (2012)
  • Author(s): Y. Hamada, T. Okaji and Y. Takei
  • Journal Title: Kyoto University Volume: 48巻 Issue: 1 Pages: 139-181
  • DOI: 10.2977/prims/65
  • Peer Reviewed
• [Journal Article] 放物型方程式に対する解の強一意接続性 (2011)
  • Author(s): 大鍛治隆司
  • Journal Title: 第50回実函数論・函数解析学合同シンポジウム講演集 Pages: 34-43
• [Journal Article] A Note on the Essential Selfadjontness of Dirac Operators with an Anomalous magnetic Moment (2011)
  • Author(s): Hubert Kalf Takashi Okaji, Osanobu Yamada
  • Journal Title: 立命館大学理工学研究所紀要 Volume: 69 Pages: 1-5
• [Presentation] Dirac 作用素に対する閾値における固有値の非存在について (2014)
  • Author(s): 大鍛治 隆司
  • Organizer: 第21回超局所解析と古典解析
  • Place of Presentation: 長崎市松藤プラザ
  • Year and Date: 2014-12-06 – 2014-12-07
• [Presentation] 2段階式Carleman評価法I,II,III (2012)
  • Author(s): 大鍛治隆司
  • Organizer: 逆問題への応用を意図した解析学の研究
  • Place of Presentation: 数理解析研究所
  • Invited
• [Presentation] note on unique continuation for parabolic operators with singular potential (2011)
  • Author(s): Takashi Okaji
  • Organizer: Perspectives in Phase Space Analysis of Partial differential equations
  • Place of Presentation: Bertinoro Italy(招待講演)
  • Year and Date: 2011-09-29
• [Presentation] 放物型方程式に対する解の強一意接続性 (2011)
  • Author(s): 大鍛治隆司
  • Organizer: 日本数学会年会
  • Place of Presentation: 早稲田大学理工学術院
  • Year and Date: 2011-03-21
• [Presentation] Spectral problems about many-body Dirac operators
  • Author(s): 大鍛治隆司
  • Organizer: スペクトル・散乱理論とその周辺
  • Place of Presentation: 数理解析研究所
  • Invited