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変分法的アプローチによる非線形楕円型方程式と拡散方程式の研究

Research Project

Project/Area Number 12J02259
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section国内
Research Field Global analysis
Research InstitutionTohoku University

Principal Investigator

生駒 典久  東北大学, 理学研究科, 助教

Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥3,630,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥330,000)
Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2013: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2012: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsWillmore 汎関数(イタリア,スイス) / Lyapunov-Schmidt 法 / Morse 理論 / 完全非線型楕円型方程式(チリ) / Pucci 作用素 / Leray-Schauder の写像度 / 非線形Kirchhoff型方程式 / 非局所効果 / 疑集解 / 最小エネルギー解 / 固有値問題 / Willmore 汎関数 / 多制限条件 / 最小化問題 / 基底状態解 / 軌道安定性 / 完全非線形楕円型作用素 / 第一固有値
Outline of Annual Research Achievements

平成26年度は3次元閉 Riemann 多様体上における Willmore 汎関数の研究および主要部が Pucci 作用素であり冪乗型の非線型反応項を伴う完全非線型楕円型方程式に関する研究を行った.
Willmore 汎関数に関する研究では,前年度までに得られていた Willmore 汎関数の展開公式を利用することにより,多様体の Ricci 曲率と scalar 曲率がある種の不等式を満たすとき,曲面積一定という制約条件を与えた下,種数が1かつ Willmore 汎関数の臨界点となる曲面(Willmore type 曲面)の存在を示した.また,Morse 理論を用いることにより,Ricci 曲率と saclar 曲率の間に課した不等式を,scalar 曲率が Morse 関数であり,その臨界点において Ricci 曲率の3つの固有値がそれぞれ相異なるという条件の下でも,種数が1である Willmore type 曲面の存在を示した.更に,Riemann 多様体の Betti 数と scalar 曲率の臨界点の情報を組み合わせることにより,種数が1である Willmore type 曲面の多重存在性に関する結果も得られた.
一方,Pucci 作用素が主要部であり,冪乗型の非線型反応項を伴う完全非線型楕円型方程式に対しては,主要部が Laplacian である場合と異なり臨界点理論が使えない.そのため新しい解析方法が必要となるが,本研究では Leray-Schauder の写像度を用いるための適切な関数空間を発見した.

Research Progress Status

26年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

26年度が最終年度であるため、記入しない。

Report

(3 results)
  • 2014 Annual Research Report
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • Research Products

    (31 results)

All 2015 2014 2013 2012

All Journal Article (5 results) (of which Peer Reviewed: 5 results,  Acknowledgement Compliant: 1 results) Presentation (26 results) (of which Invited: 15 results)

  • [Journal Article] Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials2015

    • Author(s)
      N. Ikoma
    • Journal Title

      Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

      Volume: 35 Issue: 3 Pages: 943-966

    • DOI

      10.3934/dcds.2015.35.943

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Existence and concentration result for the Kirchhoff type equatio ns with general nonlinearities2014

    • Author(s)
      G. M. Figueiredo, N. Ikoma, J. R. Santos Júnior
    • Journal Title

      Archive for Rational Mechanics and Analysis

      Volume: (印刷中)(掲載確定) Issue: 3 Pages: 931-979

    • DOI

      10.1007/s00205-014-0747-8

    • Related Report
      2014 Annual Research Report 2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Compactness of Minimizing Sequences in Nonlinear Schrodinger Systems Under Multiconstraint Conditions2014

    • Author(s)
      N. Ikoma
    • Journal Title

      Advanced Nonlinear Studies

      Volume: 14 Pages: 115-136

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence of Minimizers for Some Coupled Nonlinear Schrodinger Equations2013

    • Author(s)
      N. Ikoma
    • Journal Title

      Springer INdAM Series

      Volume: 2 Pages: 143-164

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls2012

    • Author(s)
      N. Ikoma and H. Ishii
    • Journal Title

      Annates de l'institut Henri Poincare Analyse Non Lineaire

      Volume: 29 Pages: 783-812

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Presentation] Solvabitliy of some fully nonlinear elliptic equations2015

    • Author(s)
      生駒典久
    • Organizer
      Viscosity Solutions and Related Topics
    • Place of Presentation
      早稲田大学
    • Year and Date
      2015-03-27
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 微小な曲面積を持つWillmore type torus の存在2015

    • Author(s)
      生駒典久
    • Organizer
      日本数学会2015 年度年会
    • Place of Presentation
      明治大学(駿河台)
    • Year and Date
      2015-03-21
    • Related Report
      2014 Annual Research Report
  • [Presentation] Some properties of eigenvalues of fully nonlinear elliptic operators2015

    • Author(s)
      生駒典久
    • Organizer
      第7回名古屋 微分方程式研究集会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2015-03-03
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Existence of Willmore type tori with small area in Riemannian 3-manifolds2015

    • Author(s)
      生駒典久
    • Organizer
      The 16th Northeastern Symposium on Mathematical Analysis
    • Place of Presentation
      東北大学
    • Year and Date
      2015-02-18
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Some properties of eigenvalues of fully-nonlinear elliptic operators2015

    • Author(s)
      生駒典久
    • Organizer
      第9回非線形 偏微分方程式と変分問題
    • Place of Presentation
      東京工業大学
    • Year and Date
      2015-02-08
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Existence of eigenvalues for fully nonlinear operators and their properties2015

    • Author(s)
      生駒典久
    • Organizer
      第32 回広大サステナブル科学セミナー
    • Place of Presentation
      広島大学
    • Year and Date
      2015-02-03
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Riemann 多様体上におけるWillmore torus 型曲面の存在2015

    • Author(s)
      生駒典久
    • Organizer
      金沢解析セミナー
    • Place of Presentation
      金沢大学
    • Year and Date
      2015-01-09
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Kirchhoff 型方程式に対する特異摂動問題,2015

    • Author(s)
      生駒典久
    • Organizer
      RIMS 研究集会「パターン形成と界面 ダイナミクスの数理」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2015-01-06
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Characterizations of eigenvalues for fully nonlinear operators2014

    • Author(s)
      生駒典久
    • Organizer
      RIMS 研究集会「実 領域における常微分方程式の定性的理論とその応用」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2014-11-05
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Kirchhoff 型方程式に対する特異摂動問題2014

    • Author(s)
      生駒典久
    • Organizer
      第128 回神楽坂解析セミナー
    • Place of Presentation
      東京理科大学(神楽坂)
    • Year and Date
      2014-10-25
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Kirchhoff 型方程式に対する最小エネルギー解の存在2014

    • Author(s)
      生駒典久
    • Organizer
      第3回岐阜数理科学研究会
    • Place of Presentation
      飛騨高山まちの博物館(高山市)
    • Year and Date
      2014-09-08
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Characterization of all eigenvalues of a certain class of linear and nonlinear eigenvalue problems2014

    • Author(s)
      生駒典久
    • Organizer
      Mini-workshop on Modeling, Simulation & Analysis of Pattern Formation
    • Place of Presentation
      東北大学
    • Year and Date
      2014-07-26
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Existence of least energy solutions to nonlinear Kirchhoff equations2014

    • Author(s)
      生駒典久
    • Organizer
      Seminar at SISSA
    • Place of Presentation
      SISSA (Trieste, Italy)
    • Year and Date
      2014-05-22
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Singular perturbation problems for the Kirchhoff type equations with genera 1 nonlinearities2014

    • Author(s)
      生駒 典久
    • Organizer
      日本数学会2014年度会函数方程式論分科会
    • Place of Presentation
      学習院大学(東京都豊島区)
    • Year and Date
      2014-03-17
    • Related Report
      2013 Annual Research Report
  • [Presentation] Eigenvalue problems for fully nonlinear second-order elliptic PDE on balls2014

    • Author(s)
      生駒 典久
    • Organizer
      日本数学会2014年度会函数方程式論分科会
    • Place of Presentation
      学習院大学(東京都豊島区)
    • Year and Date
      2014-03-17
    • Related Report
      2013 Annual Research Report
  • [Presentation] Kirchhoff 型方程式に対する最小エネルギー解の存在2014

    • Author(s)
      生駒 典久
    • Organizer
      第10回青葉山勉強会「変分法的手法の応用」
    • Place of Presentation
      東北大学(宮城県仙台市)
    • Year and Date
      2014-03-10
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The existence of ground state solutions to nonlinear Kirchhoff type equations with potentials2013

    • Author(s)
      生駒 典久
    • Organizer
      研究集会「微分方程式の総合的研究」
    • Place of Presentation
      東京大学(東京都目黒区)
    • Year and Date
      2013-12-22
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Singular perturbation problem of the Kirchhoff equation with general nonlinearity2013

    • Author(s)
      生駒 典久
    • Organizer
      Analysis Seminar
    • Place of Presentation
      イギリス(the University of Warwick)
    • Year and Date
      2013-06-20
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Compactness of minimizing sequences under multiconstraint conditions2013

    • Author(s)
      生駒典久
    • Organizer
      Quantization Blow-up and Concentration in Mathematical Physiscs View Point
    • Place of Presentation
      大阪大学(招待講演)
    • Year and Date
      2013-03-28
    • Related Report
      2012 Annual Research Report
  • [Presentation] Eigenvalue problems for fully nonlinear operators in intervals and balls2013

    • Author(s)
      生駒典久
    • Organizer
      Mini-WorkShop: Fully Nonlinear Equation in Valparaiso
    • Place of Presentation
      Universidad Tecnica Federico Santa Maria(チリ)(招待講演)
    • Year and Date
      2013-01-09
    • Related Report
      2012 Annual Research Report
  • [Presentation] Eigenvalue problems for fully nonlinear elliptic operators2012

    • Author(s)
      生駒典久
    • Organizer
      Swiss-Japan seminar
    • Place of Presentation
      Unlversitat Zurich (スイス)(招待講演)
    • Year and Date
      2012-12-17
    • Related Report
      2012 Annual Research Report
  • [Presentation] On behaviors of minimizing sequences related to a coupled nonlinear Schrodinger equation2012

    • Author(s)
      生駒典久
    • Organizer
      偏微分方程式セミナー
    • Place of Presentation
      北海道大学(招待講演)
    • Year and Date
      2012-11-26
    • Related Report
      2012 Annual Research Report
  • [Presentation] 完全非線形楕円型作用素に対する固有値問題2012

    • Author(s)
      生駒典久
    • Organizer
      第2回室蘭非線形解析セミナー
    • Place of Presentation
      室蘭工業大学(招待講演)
    • Year and Date
      2012-10-19
    • Related Report
      2012 Annual Research Report
  • [Presentation] 非線型連立シュレディンガー方程式系に関する最小化問題の可解性2012

    • Author(s)
      生駒典久
    • Organizer
      神戸大学解析セミナー
    • Place of Presentation
      神戸大学(招待講演)
    • Year and Date
      2012-10-18
    • Related Report
      2012 Annual Research Report
  • [Presentation] On compactness of minimizing sequences for soie nonlinear Schrodinger system2012

    • Author(s)
      生駒典久
    • Organizer
      日本数学会秋期総合分科会
    • Place of Presentation
      九州大学
    • Year and Date
      2012-09-20
    • Related Report
      2012 Annual Research Report
  • [Presentation] On behaviors of minimizing sequences for some nonlinear Schrodinger system2012

    • Author(s)
      生駒典久
    • Organizer
      Workshop on Nonlinear Dispersive PDEs
    • Place of Presentation
      東北大学(招待講演)
    • Year and Date
      2012-08-29
    • Related Report
      2012 Annual Research Report

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Published: 2013-04-25   Modified: 2024-03-26  

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