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対称臨界性原理とその非線形偏微分方程式への応用

Research Project

Project/Area Number 15654024
Research Category

Grant-in-Aid for Exploratory Research

Allocation TypeSingle-year Grants
Research Field Basic analysis
Research InstitutionWaseda University

Principal Investigator

大谷 光春  早稲田大学, 理工学術院, 教授 (30119656)

Co-Investigator(Kenkyū-buntansha) 北田 韶彦  早稲田大学, 理工学術院, 教授 (10123118)
田中 和永  早稲田大学, 理工学術院, 教授 (20188288)
松浦 啓  早稲田大学, 理工学術院, 助手 (40386628)
石渡 通徳  早稲田大学, 理工学術院, 助手 (30350458)
赤城 剛朗 (赤木 剛朗)  早稲田大学, メディアネットワークセンター, 助手 (60360202)
小林 純  早稲田大学, 理工学部, 助手 (40339700)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2005: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
KeywordsSymmetric Criticality / Subdifferential Operators / Evolution Equations / Symmetry / Critical Point / subdifferential Operators
Research Abstract

対称臨界性原理とは、「Banach空間X上で定義された汎関数Jに対し、ある群Gの作用に関して不変な部分空間X_G上でのJの臨界点が、X全体でのJの臨界点を与える」という原理である。
この原理は、Jの汎関数(フレッシェ)微分を、劣微分作用素を含むかなり一般的な多価作用素Aに置き換えても成立することが、本研究により示されている。
さらに、作用素Aが、必ずしも変分構造を有していない場合に対して拡張することが可能(AがG-共変であれば十分)であり、時間発展を伴う発展方程式に対して有用であろうことが期待されていた。
(1)非線形放物型方程式、波動方程式、シュレンディンガー方程式等への応用考えるとき、時間に関する微分作用素d/dtがどのような空間でG-共変となるかを調べることは重要であるが、今回L2(0,T;X),X=L2(Ω),L2(Ω)xL2(Ω),H10(Ω)xL2(Ω),などでそのG-共変性(G=O(N))が確かめられた。これらの空間は、上記の諸方程式を抽象発展方程式に帰着させるときに現れる基本的な空間であり、これは、今後の発展方程式への応用研究において重要な知見である。
(2)放物型方程式の時間大域解の漸近挙動を解析する際、コンパクト性は強力なな道具を提供する。一般の非有界領域では欠如しているソボレフの埋蔵定理にかかわるコンパクト性が、回転対称性を有する関数からなる部分空間においては、恢復するという事実に基づき、ある種の回転対称性を有する非有界領域におけp-Laplace作用素と爆発項を含む非線形放物型方程式の対称大域解のW1,p-有界性が確かめられた。

Report

(3 results)
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (19 results)

All 2006 2005 2004 Other

All Journal Article (12 results) Book (1 results) Publications (6 results)

  • [Journal Article] Multiple Stable Patterns for Some Reaction-Diffusion Equation in Disrupted Environments2006

    • Author(s)
      T.IDE, K.KURATA, Kazunaga TANAKA
    • Journal Title

      Discrete Contin.Dyn.Syst 14

      Pages: 93-116

    • Related Report
      2005 Annual Research Report
  • [Journal Article] L∞-energy method and its applications to some nonlinear parabolic systems2005

    • Author(s)
      Mitsuharu OTANI
    • Journal Title

      Gakuto International Series, Mathematical Sciences and Applications 22

      Pages: 233-244

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Multi-clustered high energy solutions for a phase transition problem2005

    • Author(s)
      P.FELMER, S.MARTINEZ, Kazunaga TANAKA
    • Journal Title

      Proc.Roy.Soc.Edinburgh 135A

      Pages: 731-765

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Long time behavior of solutions for a nonlinear evolution equations2005

    • Author(s)
      Kei MATSUURA
    • Journal Title

      Nonlinear Analysis 63

      Pages: 1199-1206

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On a decomposition space of a weak self similar set2005

    • Author(s)
      Akihiko KITADA, Yoshihiko OGASAWARA
    • Journal Title

      Chaos, Solitons and Fractals 24

      Pages: 785-787

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Evolution inclusions governed by subdifferentials in reflexive Banach spaces2004

    • Author(s)
      Goro AKAGI, Mitsuharu OTANI
    • Journal Title

      Journal of Evolution Equations 4

      Pages: 519-541

    • NAID

      120005372391

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Degree for subdifferential operators in Hilbert spaces2004

    • Author(s)
      Jun KOBAYASHI, Mitsuharu OTANI
    • Journal Title

      Adv.Math.Sci.Appl. 14

      Pages: 307-325

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Topological degree for S+ mappings with maximal monotone perturbations and its applications to variational inequalities2004

    • Author(s)
      Jun KOBAYASHI, Mitsuharu OTANI
    • Journal Title

      Nonlinear Anal. 59

      Pages: 147-172

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The principle of symmetric criticality for non- differentiable mappings2004

    • Author(s)
      Jun KOBAYASHI, Mitsuharu OTANI
    • Journal Title

      Journal of Functional Analysis 214

      Pages: 428-449

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities2004

    • Author(s)
      J.JEANJAEAN, Kazunaga TANAKA
    • Journal Title

      Calculus of Variations and Partial Differential Equations 21

      Pages: 287-318

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On nonstationary flows of magneto-micropolar fluid2004

    • Author(s)
      Kei MATSUURA
    • Journal Title

      Nonlinear Partial Differential Equations and Their Applications, GAKUTO International Series 20

      Pages: 480-489

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a property specific to the tent map. Chaos

    • Author(s)
      Akihiko KITADA, Yoshihiko OGASAWARA
    • Journal Title

      Solitons and Fractals (in press)

    • Related Report
      2005 Annual Research Report
  • [Book] Nonlinear Partial Differential Equations and Their Applications2004

    • Author(s)
      N.KENMUUHI, Mitsuharu OTANI, S.Zheng
    • Total Pages
      534
    • Publisher
      GAKKOTOSHO, International Series vol.20
    • Related Report
      2004 Annual Research Report
  • [Publications] G.Akagi, Jun Kobayashi, M.Otani: "Principle of symmetric criticality and evolution equations"Dynamical Systems and Differential Equations edited by W.Fei, S.Hu & X.Lin, American Institute of Mathematical Sciences. 1-10 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] G.Akagi, M.Otani: "Evolution equations and subdifferentials in Banach spaces"Dynamical Systems and Differential Equations edited by W.Fei, S.Hu & X.Lin, American Institute of Mathematical Sciences. 11-20 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Garcia-Huidobro, R.Manasevich, M.Otani: "Existence results for p-Laplacian-like systems od O.E.D.'s"Funkcialaj Ekvacioj. 46. 253-285 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Kitada: "A note on a property of a weak self-similar perfect set"Chaos Solitons Fractals. 15. 903-907 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Ding, K.Tanaka: "Multiplicity of positive solutions of a nonlinear Schrodinger equation"Manuscripta Mathematica. 112. 109-135 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] L.Jeanjean, K.Tanaka: "A remark on least energy solutions in R^N"Proc.Amer.Math.Soc. 131. 2399-2408 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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