Spectral analysis of Neumann-Poincare operators and its applications

• Project/Area Number: 17K05303
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Ehime University
• Principal Investigator: ANDO Kazunori 愛媛大学, 理工学研究科(工学系), 准教授 (70774884)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: スペクトル解析 / ノイマン-ポアンカレ作用素 / プラズモン共鳴 / スペクトル理論 / 散乱理論 / レゾナンス / 解析学 / 数理物理 / 応用数学

Outline of Final Research Achievements

We have studied the relationship between the eigenvalues of Neumann-Poincare operators which are boundary integral operators, and the smoothness of the boundary, mainly in two-dimansions. The decay rate of the eigenvalues of Neumann-Poincare operator has a close relationship with anomalous localized resonance. We have studied the Neumann-Poincare operator associated to both the Laplace operator and the Lame operator, and we obtained the followings: if the boundary is an analytic curve, then the eigenvalues of Neumann-Poincare operator accumulate exponentially.

Academic Significance and Societal Importance of the Research Achievements

異常局在共鳴とは、数学的には微分方程式の楕円性が崩れる際にある条件のもとに見られる現象で、領域外部にエネルギーの供給源がある場合に領域境界上でエネルギーの異常な高まりを示し、遠方では微分方程式の解が有界となることである。異常局在共鳴は、物理的にも大変興味深い現象でクローキングやスーパーレンズなど、さまざまな応用が考えられている。異常局在共鳴を示す際には、ノイマン-ポアンカレ作用素の固有値の集積の速さが密接に関係していることが知られており、今後の研究でどのような性質をもった領域で異常局在共鳴が起こるかを研究する際の足掛かりとなる研究である。

Research Products

• [Int'l Joint Research] Inha University(韓国)
• [Journal Article] Spectral analysis of Neumann-Poincare operator (2021)
  • Author(s): Kazunori Ando, Hyeonbae Kang, Yoshihisa Miyanishi, Mihai Putinar
  • Journal Title: ROMANIAN JOURNAL OF PURE AND APPLIED MATHEMATICS Volume: -
  • Int'l Joint Research
• [Journal Article] The First Hadamard Variation of Neumann-Poincare eigenvalues on the sphere (2019)
  • Author(s): Kazunori Ando, Yoshihisa Miyanishi, Hyeonbae Kang and Erika Ushikoshi
  • Journal Title: Proceedings of the American Mathematical Society Volume: 147 Issue: 3 Pages: 1073-1080
  • DOI: 10.1090/proc/14246
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Elastic Neumann-Poincare Operators on Three Dimensional Smooth Domains: Polynomial Compactness and Spectral Structure (2019)
  • Author(s): Kazunori Ando, Hyeonbae Kang, Yoshihisa Miyanishi
  • Journal Title: INTERNATIONAL MATHEMATICS RESEARCH NOTICES Volume: 2019 Issue: 12 Pages: 3883-3900
  • DOI: 10.1093/imrn/rnx258
  • Peer Reviewed
• [Journal Article] Exponential decay estimates of the eigenvalues for the Neumann-Poincare operator on analytic boundaries in two dimensions (2018)
  • Author(s): Kazunori Ando, Yoshihisa Miyanishi and Hyeonbae Kang
  • Journal Title: Journal of Integral Equations and Applications Volume: 30 Issue: 4 Pages: 473-489
  • DOI: 10.1216/jie-2018-30-4-473
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Eigenvalues of the Neumann-Poincare operator in two dimensions (2020)
  • Author(s): 安藤和典
  • Organizer: 微分方程式の総合的研究
• [Presentation] Spectral properties of the Neumann-Poincaré operators and anomalous localized resonance (2020)
  • Author(s): Kazunori Ando
  • Organizer: スペクトル・散乱 那覇シンポジウム
  • Int'l Joint Research / Invited
• [Presentation] Eigenvalues of the Neumann-Poincaré operators in two dimensions (2019)
  • Author(s): Kazunori Ando
  • Organizer: The 9th Congress of Romanian Mathematicians
  • Int'l Joint Research / Invited
• [Presentation] The eigenvalues of the elastic Neumann-Poincaré operator in two dimensions (2019)
  • Author(s): Kazunori Ando
  • Organizer: Mini-workshop: Neumann-Poincaré operator and related topics
  • Int'l Joint Research
• [Presentation] The eigenvalues of the elastic Neumann-Poincare operator in two dimensions (2019)
  • Author(s): 安藤和典
  • Organizer: Mini-workshop: Neumann-Poincare operator and related topics
  • Int'l Joint Research
• [Presentation] Spectral analysis of the NP operator in two dimensions and cloaking by anomalous localized resonance (2018)
  • Author(s): Kazunori Ando
  • Organizer: International Workshop: The Neumann-Poincare Operator, Plasmonics, and Field Concentration
  • Int'l Joint Research / Invited
• [Presentation] Spectral analysis of the elastic Neumann-Poincare operator and cloaking by anomalous localized resonance (2018)
  • Author(s): Kazunori Ando
  • Organizer: The 3rd East Asia Section of IPIA Young Scholars Symposium
  • Int'l Joint Research / Invited
• [Presentation] 弾性方程式に対するノイマン-ポアンカレ作用素のスペクトルについて (2017)
  • Author(s): 安藤和典
  • Organizer: 夏の作用素論シンポジウム