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Asymptotic analysis on solutions to weakly coupled systems for fully nonlinear equations

Research Project

Project/Area Number 24840043
Research Category

Grant-in-Aid for Research Activity Start-up

Allocation TypeSingle-year Grants
Research Field Basic analysis
Research InstitutionHiroshima University (2013)
Fukuoka University (2012)

Principal Investigator

MITAKE Hiroyoshi  広島大学, サステナブル・ディベロップメント実践研究センター, 特任講師 (90631979)

Project Period (FY) 2012-08-31 – 2014-03-31
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywordsハミルトン・ヤコビ方程式 / 弱結合型連立方程式 / 長時間挙動 / 均質化問題 / 退化粘性ハミルトン・ヤコビ方程式 / 非線形随伴法 / 障害問題 / 無限大ラプラス方程式 / 均質化理論 / 弱KAM理論 / 最適切替コスト問題
Research Abstract

The main purpose is to deeply understand the structure of solutions to the weakly coupled system of fully nonlinear equations. More precisely, I mainly focused on some asymptotic problems (homogenization and large-time behavior), and got new results as follows. 1. Homogenization of the weakly coupled system of Hamilton-Jacobi equation with fast switching rates, analysis on the initial layer and the effective Hamiltonian. 2. Large-time behavior of the same system by analyzing the value functions of the optimal control problems which appear in the dynamic programming for the system whose states are governed by random changes. 3. Large-time asymptotics for degenerate viscous Hamilton-Jacobi equations by using the nonlinear adjoint method, which was completely new and giving a breakthrough on the study of the large-time asymptotics for degenerate parabolic equations.

Report

(3 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Annual Research Report
  • Research Products

    (43 results)

All 2014 2013 2012 Other

All Journal Article (21 results) (of which Peer Reviewed: 21 results,  Open Access: 1 results) Presentation (21 results) (of which Invited: 11 results) Remarks (1 results)

  • [Journal Article] Homogenization of weakly coupled systems of Hamilton-Jacobi equations with fast switching rates2014

    • Author(s)
      H. Mitake, H. V. Tran
    • Journal Title

      Arch. Ration. Mech. Anal

      Volume: 211, no. 3 Issue: 3 Pages: 733-769

    • DOI

      10.1007/s00205-013-0685-x

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton-Jacobi equations2014

    • Author(s)
      H. Mitake, H. V. Tran
    • Journal Title

      J. Math. Pures Appl

      Volume: (9) 101, no. 1 Issue: 1 Pages: 76-93

    • DOI

      10.1016/j.matpur.2013.05.004

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Singular Neumann problems and large-time behavior of solutions of noncoercive Hamilton-Jacobi equations2014

    • Author(s)
      Y. Giga, Q. Liu and H. Mitake
    • Journal Title

      Trans. Amer. Math. Soc.

      Volume: 366 Issue: 4 Pages: 1905-1941

    • DOI

      10.1090/s0002-9947-2013-05905-3

    • NAID

      120006462288

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Singular Neumann problems and large-time behavior of solutions of non-coercive Hamilton-Jacobi equations2014

    • Author(s)
      Y. Giga, Q. Liu and H. Mitake
    • Journal Title

      Transactions of the American Mathematical Society

      Volume: 366, no. 4 Pages: 1905-1941

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton-Jacobi equations2014

    • Author(s)
      H. Mitake and H. V. Tran
    • Journal Title

      Journal de Mathematiques Pures et Appliquees

      Volume: (9) 101, no. 1 Pages: 76-93

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Homogenization of weakly coupled systems of Hamilton-Jacobi equations with fast switching rates2014

    • Author(s)
      H. Mitake and H. V. Tran
    • Journal Title

      Archive for Rational Mechanics and Analysis

      Volume: 211, no. 3, Pages: 733-769

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians2014

    • Author(s)
      F. Cagnetti, D. A. Gomes, H. Mitake, and H. V. Tran
    • Journal Title

      Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

      Volume: 印刷中

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Large-time asymptotics for Hamilton-Jacobi equations with noncoercive Hamiltonians appearing in crystal growth2014

    • Author(s)
      Y. Giga, Q. Liu, and H. Mitake
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 印刷中

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A new PDE approach to the large time asymptotics of solutions of Hamilton.Jacobi equations2013

    • Author(s)
      G.Barles, H.Ishii and H.Mitake
    • Journal Title

      Bull. Math. Sci

      Volume: (in press) Issue: 3 Pages: 363-388

    • DOI

      10.1007/s13373-013-0036-0

    • Related Report
      2013 Final Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations2013

    • Author(s)
      G. Barles, H. Ishii and H. Mitake
    • Journal Title

      Bulletin of Mathematical Sciences

      Volume: 3, no. 3 Pages: 363-388

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Singular Neumann problems and large-time behavior of solutions of non-coercive Hamilton-Jacobi equations2013

    • Author(s)
      Yoshikazu Giga, Qing Liu, Hiroyoshi Mitake
    • Journal Title

      Transactions of the American Mathematical Society

      Volume: 掲載確定

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton-Jacobi equations2013

    • Author(s)
      Hiroyoshi Mitake, Hung V. Tran
    • Journal Title

      Journal de Mathematiques Pures et Appliquees

      Volume: 掲載確定

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations2013

    • Author(s)
      Guy Barles, Hitoshi Ishii, Hiroyoshi Mitake
    • Journal Title

      Bulletin of Mathematical Sciences

      Volume: 掲載確定

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Large-time asymptotics for Hamilton-Jacobi equations with noncoercive Hamiltonians appearing in crystal growth2013

    • Author(s)
      Yoshikazu Giga, Qing Liu, Hiroyoshi Mitake
    • Journal Title

      Proceeding of the 4th MSJ-SI Conference

      Volume: 掲載確定

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton-Jacobi equations2012

    • Author(s)
      H. Mitake, H. V. Tran
    • Journal Title

      Asymptot. Anal

      Volume: Vol. 77 Issue: 1-2 Pages: 43-70

    • DOI

      10.3233/asy-2011-1071

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the large time behavior of solutions of Hamilton-Jacobi equations associated with nonlinear boundary conditions2012

    • Author(s)
      Guy Barles, 石井仁司, 三竹大寿
    • Journal Title

      Archive Rational Mechanic Analysis

      Volume: (印刷中) Issue: 2 Pages: 515-558

    • DOI

      10.1007/s00205-011-0484-1

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Large-time asymptotics for one-dimensional Dirichlet problems for Hamilton-Jacobi equations with non-coercive Hamiltonians2012

    • Author(s)
      儀我美一, Qing Liu, 三竹大寿
    • Journal Title

      Journal of Differential Equations

      Volume: 252 Issue: 2 Pages: 1263-1282

    • DOI

      10.1016/j.jde.2011.10.010

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton-Jacobi equations2012

    • Author(s)
      Guy Barles, 三竹大寿
    • Journal Title

      Communications Partial Differential Equations

      Volume: 37 Issue: 1 Pages: 136-168

    • DOI

      10.1080/03605302.2011.553645

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations2012

    • Author(s)
      Guy Barles, Olivier Ley, 三竹大寿
    • Journal Title

      Mathematische Annalen

      Volume: 352 Issue: 2 Pages: 409-451

    • DOI

      10.1007/s00208-011-0648-1

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Large-time asymptotics for Hamilton-Jacobi equations with noncoercive Hamiltonians appearing in crystal growth

    • Author(s)
      Y. Giga, Q. Liu, H. Mitake
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: (印刷中)

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians

    • Author(s)
      F. Cagnetti, D. A. Gomes, H. Mitake, H. V. Tran
    • Journal Title

      Inst. H. Poincare Anal. Non Lineaire

      Volume: (印刷中)(to appear in Ann)

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Presentation] Weakly coupled systems of the infinity Laplace equations2014

    • Author(s)
      Hiroyoshi Mitake
    • Organizer
      AMCS seminar in KAUST
    • Place of Presentation
      Saudi Arabia
    • Year and Date
      2014-03-10
    • Related Report
      2013 Final Research Report
  • [Presentation] Large time behavior for degenerate viscous Hamilton-Jacobi equa- tions by the nonlinear adjoint method2014

    • Author(s)
      三竹大寿
    • Organizer
      第31回九州における偏微分方程式研究集会
    • Place of Presentation
      福岡
    • Year and Date
      2014-01-29
    • Related Report
      2013 Final Research Report
  • [Presentation] Large-time asymptotics for degenerate viscous Hamilton-Jacobi equations2013

    • Author(s)
      Hiroyoshi Mitake
    • Organizer
      Colloquium
    • Place of Presentation
      Kansas, U.S.A.
    • Year and Date
      2013-10-10
    • Related Report
      2013 Final Research Report
  • [Presentation] Homogenization of weakly coupled systems of Hamilton-Jacobi equations with fast switching rates2013

    • Author(s)
      Hiroyoshi Mitake, H. V. Tran
    • Organizer
      AMS conference
    • Place of Presentation
      Louisville, U.S.A.
    • Year and Date
      2013-10-06
    • Related Report
      2013 Final Research Report
  • [Presentation] A dynamical approach to the large time asymptotics for weakly coupled systems of Hamilton-Jacobi equations2013

    • Author(s)
      三竹大寿
    • Organizer
      Computational, Applied Mathematics and PDE seminar
    • Place of Presentation
      Chicago, U.S.A.
    • Year and Date
      2013-03-06
    • Related Report
      2013 Final Research Report
  • [Presentation] A new method for large-time asymptotics for degenerate viscous Hamilton-Jacobi equations by the nonlinear adjoint technique2013

    • Author(s)
      三竹大寿
    • Organizer
      Fukae workshop on PDE (2013)
    • Place of Presentation
      神戸
    • Year and Date
      2013-01-25
    • Related Report
      2013 Final Research Report
  • [Presentation] ハミルトン・ヤコビ方程式の弱結合型システムに関する長時間挙動:力学的アプローチ

    • Author(s)
      三竹 大寿, H. V. Tran
    • Organizer
      日本数学会
    • Place of Presentation
      愛媛大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] 非線形随伴法を用いた長時間挙動に関する解析: 半線形退化放物型偏微分方程式

    • Author(s)
      三竹 大寿, F. Cagnetti, D. A. Gomes, H. V. Tran
    • Organizer
      日本数学会
    • Place of Presentation
      愛媛大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] Homogenization of weakly coupled systems of Hamilton-Jacobi equations with fast switching rates

    • Author(s)
      H. Mitake, H. V. Tran
    • Organizer
      AMS conference
    • Place of Presentation
      Louisville, USA
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Large-time asymptotics for degenerate viscous Hamilton-Jacobi equations

    • Author(s)
      H. Mitake
    • Organizer
      Colloquium in Kansas University
    • Place of Presentation
      Kansas, USA
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Hamilton-Jacobi 方程式の解の長時間挙動ー非線形随伴法を用いた解析ー

    • Author(s)
      三竹 大寿
    • Organizer
      京都駅前セミナー
    • Place of Presentation
      京都
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Large time behavior for degenerate viscous Hamilton-Jacobi equations by the nonlinear adjoint method

    • Author(s)
      H. Mitake
    • Organizer
      第31 回九州における偏微分方程式研究集会
    • Place of Presentation
      福岡大学
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Weakly coupled systems of the infinity Laplace equations

    • Author(s)
      H. Mitake
    • Organizer
      AMCS seminar in KAUST
    • Place of Presentation
      Saudi Arabia
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton-Jacobi equations

    • Author(s)
      三竹 大寿
    • Organizer
      第八回非線型の諸問題
    • Place of Presentation
      宮崎
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] ハミルトン・ヤコビ方程式に対する非線形ノイマン型境界値問題の解の長時間挙動

    • Author(s)
      三竹 大寿
    • Organizer
      2012年度秋季総合分科会
    • Place of Presentation
      九州大学
    • Related Report
      2012 Annual Research Report
  • [Presentation] ハミルトン・ヤコビ方程式の弱結合型システムに関する漸近問題

    • Author(s)
      三竹 大寿
    • Organizer
      2012年度秋季総合分科会
    • Place of Presentation
      九州大学
    • Related Report
      2012 Annual Research Report
  • [Presentation] ファセット結晶形状の不安定性と非強圧的ハミルトン・ヤコビ方程式

    • Author(s)
      三竹 大寿
    • Organizer
      表面・界面ダイナミクスの数理IV
    • Place of Presentation
      東京大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] A dynamical approach to the large time asymptotics for weakly coupled systems of Hamilton-Jacobi equations

    • Author(s)
      三竹 大寿
    • Organizer
      熊本大学応用解析セミナー
    • Place of Presentation
      熊本大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] A dynamical approach to the large time asymptotics for weakly coupled systems of Hamilton-Jacobi equations

    • Author(s)
      三竹 大寿
    • Organizer
      FMSP Tutorial Workshop on Weak KAM Theory and Related Topics
    • Place of Presentation
      東京大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] A new method for large-time asymptotics for degenerate viscous Hamilton-Jacobi equations by the nonlinear adjoint technique

    • Author(s)
      三竹 大寿
    • Organizer
      Fukae workshop on PDE (2013)
    • Place of Presentation
      神戸大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] A dynamical approach to the large time asymptotics for weakly coupled systems of Hamilton-Jacobi equations

    • Author(s)
      三竹 大寿
    • Organizer
      Computational, Applied Mathematics and PDE seminar
    • Place of Presentation
      The University of Chicago, アメリカ
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Remarks] 福岡大学研究者情報

    • URL

      http://resweb2.jhk.adm.fukuoka-u.ac.jp/FukuokaUnivHtml/info/5904/R108J.html?P=Mon%20Apr%2029%202013%2018:22:43%20GMT+0900%20(JST)

    • Related Report
      2012 Annual Research Report

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Published: 2012-11-27   Modified: 2019-07-29  

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