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2013 Fiscal Year Final Research Report

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

Research Project

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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
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
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.

  • Research Products

    (17 results)

All 2014 2013 2012 Other

All Journal Article (11 results) (of which Peer Reviewed: 10 results) Presentation (6 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 Pages: pp. 733-769

    • DOI

      10.1007/s00205-013-0685-x

    • 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 Pages: pp. 76-93

    • DOI

      10.1016/j.matpur.2013.05.004

    • 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, H. Mitake
    • Journal Title

      Trans. Amer. Math. Soc

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

    • DOI

      10.1090/S0002-9947-2013-05905-3

  • [Journal Article] A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations2013

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

      Bull. Math. Sci

      Volume: 3, no. 3 Pages: pp. 363-388

    • DOI

      10.1007/s13373-013-0036-0

    • 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 Pages: pp. 43-70

    • DOI

      10.3233/ASY-2011-1071

    • Peer Reviewed
  • [Journal Article] On the large time behavior of solutions of Hamilton-Jacobi equations associated with nonlinear boundary conditions2012

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

      Arch. Ration. Mech. Anal

      Volume: Vol. 204 Pages: pp. 515-558

    • DOI

      10.1007/s00205-011-0484-1

    • Peer Reviewed
  • [Journal Article] Large-time asymptotics for one-dimensional Dirichlet problems for Hamilton-Jacobi equations with non-coercive Hamiltonians2012

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

      J. Differential Equations

      Volume: Vol. 252 Pages: pp. 1263-1282

    • DOI

      10.1016/j.jde.2011.10.010

    • Peer Reviewed
  • [Journal Article] A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton-Jacobi equations2012

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

      Comm. Partial Differential Equations

      Volume: Vol. 37 Pages: pp. 136-168

    • DOI

      10.1080/03605302.2011.553645

    • Peer Reviewed
  • [Journal Article] Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations2012

    • Author(s)
      G. Barles, O. Ley, H. Mitake
    • Journal Title

      Math. Ann

      Volume: Vol. 352 Pages: pp. 409-451

    • DOI

      10.1007/s00208-011-0648-1

    • 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)

    • 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)

    • 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
  • [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
  • [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
  • [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
  • [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
  • [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

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Published: 2015-07-16  

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