Spectral analysis of elliptic PDE system in complicated domains and application

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03576
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Hokkaido University
• Principal Investigator: JIMBO Shuichi 北海道大学, 理学研究院, 特任教授 (80201565)
• Co-Investigator(Kenkyū-buntansha): 本多 尚文 北海道大学, 理学研究院, 教授 (00238817)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: ラメ方程式系 / ストークス方程式系 / スペクトル問題 / 特異領域変形

Outline of Final Research Achievements

1. We studied the spectral problem of the Lame system which is a model equation of deformation or osscillation of an elastic body. I studied the case that the body has a small hole or a thin tunnel and obtained a perturbation formula of each eigenvalue. In that process of research we considered the boundary value problem of the homogeneous Lame system in a 2d annulus and got an explicit expression of the solution which takes an infinite sequence of fundamental solution basis. Subsequently we studied the 3dimensional body with a thin tunnel and obtained a similar formula.
2. Similarly to the item 1, we studied the same spectral problem for the Stokes system and obtained the corresponding results.

Free Research Field

偏微分方程式, 応用解析学

Academic Significance and Societal Importance of the Research Achievements

ラプラシアン等の２階楕円型作用素では同様の先行の研究結果は知られていたが, ラメの方程式系やストークスの方程式系ではこのような結果は得られていなかったので, このような物理的背景をもつ偏微分方程式の領域依存性に関する理解の深化に貢献している. また工学的あるいは物理的な背景もつ成果なので諸科学の分野の基礎を固めることにもつながっている.