Singular domain deformation and analysis on elliptic operators in elasticity and electromagnetism

• Project/Area Number: 25400153
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Hokkaido University
• Principal Investigator: Jimbo Shuichi 北海道大学, 理学(系)研究科(研究院), 教授 (80201565)
• Co-Investigator(Kenkyū-buntansha): HONDA NAOFUMI 北海道大学, 大学院理学研究院, 教授 (00238817) ; TONEGAWA YOSHIHIRO 東京工業大学, 理工学研究科, 教授 (80296748)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 特異的領域変形 / 楕円型作用素 / 固有値問題 / ラプラス作用素 / ストークス作用素 / ラメ作用素 / スペクトル / マックスウェルの作用素 / 固有振動 / 摂動公式 / 領域変形

Outline of Final Research Achievements

I studied spectra of elliptic operators for regularly or singularly deformed domain (Lame operator, Stokes operator, Maxwell operator). (i) I studied polynomial solutions, rational type solutions with their structures of homogeneous Stokes and Elastic equations (with H.Ito, N.Honda), (ii) I obtained spectral Hadamard variational formula of Stokes operator, Maxwell operator for regularly perturbed domain for Dirichlet and Slip type boundary condition (with E. Ushikoshi). I obtained an elaborate behaviors of eigenvalues for Maxwell operator, (iii) I studied elaborate behaviors of eigenvalues of Lame or Maxwell operators in a domain with small hole, (iv) I obtained elaborate behaviors of eigenfrequencies of elastic body composed of several thin rod.

Research Products

• [Journal Article] Hadamard variational formula for the eigenvalue of the Stokes equations with the Dirichlet boundary conditions (2015)
  • Author(s): S. Jimbo, E. Ushikoshi
  • Journal Title: Far East Journal of Mathematical Sciences Volume: 98 Pages: 713-739
  • Peer Reviewed
• [Journal Article] Eigenvalues of the Laplacian in a domain with a thin tubular hole (2015)
  • Author(s): S. Jimbo
  • Journal Title: Journal of Elliptic and Parabolic Equations Volume: 1 Pages: 137-174
  • Peer Reviewed
• [Journal Article] Spectral comparison and gradient-like property in the Fitz-Nagumo type Equations (2015)
  • Author(s): C.N. Chen, S. Jimbo, Y. Morita
  • Journal Title: Nonlinearity Volume: 28 Pages: 1003-1016
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] On the geometric aspect of Sato's postulates on the S-matrix (2015)
  • Author(s): N. Honda, T. Kawai, H.P. Stapp
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B52 Pages: 11-54
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Convergence of the Allen-Cahn equation with Neumann boundary conditions (2015)
  • Author(s): M. Mizuno, Y. Tonegawa
  • Journal Title: SIAM Journal on Mathematical Analysis Volume: 47 Pages: 1906-1932
  • NAID: 120006459746
  • Peer Reviewed
• [Journal Article] Spectral comparison and gradient-like property (2015)
  • Author(s): Chao-Nien Chen, Shuichi Jimbo, and Yoshihisa Morita
  • Journal Title: Nonlinearity Volume: 28 Issue: 4 Pages: 1003-1016
  • DOI: 10.1088/0951-7715/28/4/1003
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Hadamard Variation for Electromagnetic Frequencies (2013)
  • Author(s): Shuichi JIMBO
  • Journal Title: Geometric Properties for Parabolic and Elliptic PDE's, Volume: 2 Pages: 179-199
  • Peer Reviewed
• [Journal Article] Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation (2013)
  • Author(s): Shuichi Jimbo, Yoshihisa Morita
  • Journal Title: J. Differential Equations Volume: 255 Pages: 1657-1683
  • Peer Reviewed
• [Presentation] Reaction diffusion equations in an infinite Y-shaped Graph and a time entire solutions (2016)
  • Author(s): Shuichi Jimbo
  • Organizer: Geometry of PDEs and its related Problems
  • Place of Presentation: Tohoku University, Miyagi prefecture, Sendai city
  • Year and Date: 2016-10-06
  • Int'l Joint Research / Invited
• [Presentation] 特異的領域変形と固有値問題 (2015)
  • Author(s): 神保秀一
  • Organizer: パターン形成とダイナミクスの解構造の探求
  • Place of Presentation: 学術交流会館, 北海道大学, 北海道, 札幌市
  • Year and Date: 2015-06-26
  • Invited
• [Presentation] Spectra of domains with a thin tubular hole (2015)
  • Author(s): Shuichi Jimbo
  • Organizer: Colloquim
  • Place of Presentation: Inha University, Incheon, Korea
  • Year and Date: 2015-04-28
  • Invited
• [Presentation] Eigenvalues of the Laplacian in a domain with a thin tubular hole (2014)
  • Author(s): Shuichi JIMBO
  • Organizer: 2014NCTS Workshop on Calculus of Variations and Related topics
  • Place of Presentation: NCTS National Tsing-Hua Univ.（台湾）
  • Year and Date: 2014-10-29 – 2014-10-31
  • Invited
• [Presentation] Eigenvalues of the Laplacian in a domain with a thin tubular hole (2014)
  • Author(s): Shuichi JIMBO
  • Organizer: 8th European Conference on Elliptic and Parabolic Problems
  • Place of Presentation: Hotel Serapo(Gaeta, Italy)
  • Year and Date: 2014-05-26 – 2014-05-30
  • Invited
• [Presentation] Eigenvalues of Laplacian in a domain with a thin tubular hole (2013)
  • Author(s): Shuichi JIMBO
  • Organizer: 日本数学会, 応用数学分科会
  • Place of Presentation: 愛媛大学, 松山市