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Studies on variational problems, optimization problems and nonlinear partial differential equations

Research Project

Project/Area Number 19K03587
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionTokyo Metropolitan University

Principal Investigator

Kurata Kazuhiro  東京都立大学, 理学研究科, 教授 (10186489)

Project Period (FY) 2019-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords変分問題 / パターン形成 / 非線形シュレディンガー方程式 / 逆問題 / 凝集現象 / 漸近挙動 / メトリックグラフ / エネルギー最小解 / 非一様性 / パタン形成 / 偏微分方程式 / 固有値問題 / 漸近解析
Outline of Research at the Start

量子物理現象、化学反応現象や生命現象を含むさまざまな非線形現象の数理モデルとして現れる非線形偏微分方程式の解の構造の厳密な数学解析を通して、数学的視点からの現象のより深い理解を獲得することをめざす。特に、境界条件と環境効果の解の構造への影響、定常パターン形成のメカニズムと形状、固有値最適化問題に関するより深い理解を変分法、スペクトル解析および漸近解析を用いて推進していく。

Outline of Final Research Achievements

In this study, we treated several nonlinear elliptic boundary value problems associated with corresponding pattern formation phenomena, for example FitzHugh-Nagumo system, Schanekenberg model, Keller-Segel chemotaxis model. Moreover, we also studied an inverse boundary value problem for the magnetic Shroedinger equation and several nonlinear elliptic problem on compact metric graphs.
We obtained results to clarify the relationship between the network structure of the compact metric graphs and the structure of solutions.

Academic Significance and Societal Importance of the Research Achievements

様々なパターン形成問題の定常パターン形成のメカニズムを非線形楕円型微分方程式で記述される数理モデルの精密な数学解析を通して、理解することができるという点で、学術的意義は深いと考えている。様々な複雑な自然現象の基本的なメカニズムが単純な数理モデルに内在することを示しているという点で、数学解析の持つ社会的意義は大きいと思われる。

Report

(6 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (22 results)

All 2024 2023 2022 2021 2020 2019 Other

All Journal Article (8 results) (of which Peer Reviewed: 7 results) Presentation (10 results) (of which Invited: 8 results) Book (1 results) Remarks (3 results)

  • [Journal Article] Variational problems associated with a system of nonlinear Schroedinger equations with three wave interaction2022

    • Author(s)
      Kazuhiro Kurata and Yuki Osada
    • Journal Title

      Disc, Conti. Dyn. Syst.

      Volume: 27 Issue: 3 Pages: 1511-1547

    • DOI

      10.3934/dcdsb.2021100

    • Related Report
      2022 Research-status Report 2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Reconstruction of the Defect by the Enclosure Method for Inverse Problems of the Magnetic Schroedinger Operators2022

    • Author(s)
      Kazuhiro Kurata and Ryusei Yamashita
    • Journal Title

      Tokyo J. Mathematics

      Volume: 45 Issue: 2 Pages: 547-577

    • DOI

      10.3836/tjm/1502179363

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs2021

    • Author(s)
      Ishii Yuta、Kurata Kazuhiro
    • Journal Title

      Communications on Pure & Applied Analysis

      Volume: 20 Issue: 4 Pages: 1633-1679

    • DOI

      10.3934/cpaa.2021035

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Asymptotic expansion of the ground state energy for nonlinear Schroedinger system with three wave interaction.2021

    • Author(s)
      Kauhiro Kurata and Yuki Osada
    • Journal Title

      Comm. Pure Appl. Anal.

      Volume: 20 Issue: 12 Pages: 4239-4251

    • DOI

      10.3934/cpaa.2021157

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Least energy solutions to semi-linear elliptic problems on metric graphs2020

    • Author(s)
      Kurata Kazuhiro, Shibata Masataka
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 491 Issue: 1 Pages: 124297-124297

    • DOI

      10.1016/j.jmaa.2020.124297

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Construction and stability analysis of one-peak symmetric stationary solutions to the Schnakenberg model with heterogeneity2020

    • Author(s)
      Yuta Ishii and Kazuhiro Kurata
    • Journal Title

      RIMS Kokyuroku

      Volume: 2146 Pages: 03-116

    • NAID

      120006888290

    • Related Report
      2019 Research-status Report
  • [Journal Article] Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity2019

    • Author(s)
      Yuta Ishii and Kazuhiro Kurata
    • Journal Title

      Discrete Contin. Dyn. Syst.

      Volume: 39 Pages: 2807-2875

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] A Singular Perturbation Problem for heteroclinic Solutions to the FitzHugh-Nagumo Type Reaction-Diffusion System with heterogeneity2019

    • Author(s)
      Takashi Kajiwara and Kazuhiro Kurata
    • Journal Title

      J. Math. Sci. Univ. Tokyo

      Volume: 26 Pages: 141-199

    • NAID

      120006872366

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Presentation] Green functions, Keller=Segel's chemotaxis model and eigenvalue optimization problems on compact metric graphs2024

    • Author(s)
      倉田 和浩
    • Organizer
      第13回室蘭非線形解析研究会
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Nonlinear elliptic singular perturbation problems on compact metric graphs I,II2023

    • Author(s)
      倉田 和浩
    • Organizer
      第9回偏微分方程式レクチャーシリーズin福岡工業大学
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Concentration phenomena of Keller-Segel's minimal chemotaxis model on compact metric graphs2023

    • Author(s)
      倉田 和浩
    • Organizer
      RIMS「常微分方程式の定性的理論の発展とその応用」
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] On some variational problem related to a reaction-diffusion system with mass conservation on a metric graph2023

    • Author(s)
      Kazuhiro Kurata
    • Organizer
      Workshop on Variational Methods and Functional Inequalities (OCAMI)
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] 空間非一様な係数を含む交差拡散数理モデルにおけるシャドウ系とその解析2021

    • Author(s)
      倉田 和浩
    • Organizer
      RIMS研究集会
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] メトリックグラフ上のいくつかの特異極限問題の解の存在と形状について2021

    • Author(s)
      倉田 和浩
    • Organizer
      早稲田大学応用解析研究会
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Location of spikes of spiky stationary solutions to some nonlinear elliptic problems on compact metric graphs2020

    • Author(s)
      Kazuhiro Kurata
    • Organizer
      「微分方程式の総合的研究」
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] 3波相互作用をもつ非線形シュレディンガー方程式系の特異摂動問題2020

    • Author(s)
      Kazuhiro Kurata
    • Organizer
      「第14回非線形偏微分方程式と変分問題」
    • Related Report
      2019 Research-status Report
  • [Presentation] On an optimization problem of the first eigenvalue of the Laplacian on a thin domain with Neumann window2019

    • Author(s)
      Kazuhiro Kurata
    • Organizer
      「解析学特別研究セミナー」
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] compact metric graph上のある特異摂動問題の凝縮解について2019

    • Author(s)
      Kazuhiro Kurata
    • Organizer
      「第29回数理物理と微分方程式」
    • Related Report
      2019 Research-status Report
  • [Book] フーリエ解析の基礎と応用2020

    • Author(s)
      倉田 和浩
    • Total Pages
      227
    • Publisher
      数理工学社
    • ISBN
      9784864810678
    • Related Report
      2020 Research-status Report
  • [Remarks] Kurata's Home Page

    • URL

      https://www.comp.tmu.ac.jp/tmu-kurata/index.html

    • Related Report
      2020 Research-status Report
  • [Remarks] Kurata's Home Page

    • URL

      http://www.comp.tmu.ac.jp/tmu-kurata/index.html

    • Related Report
      2019 Research-status Report
  • [Remarks] 教員紹介:倉田 和浩 | 東京都立大学

    • URL

      https://www.tmu.ac.jp/stafflist/data/ka/450.html

    • Related Report
      2019 Research-status Report

URL: 

Published: 2019-04-18   Modified: 2025-01-30  

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