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Research on viscosity solutions of differential equations and their applications

Research Project

Project/Area Number 12440044
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionWaseda University (2001-2002)
Tokyo Metropolitan University (2000)

Principal Investigator

ISHII Hitoshi  Waseda University, School of Education, Professor, 教育学部, 教授 (70102887)

Co-Investigator(Kenkyū-buntansha) SAKAI Makoto  Tokyo Metropolitan University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70016129)
MOCHIZUKI Kiyoshi  Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (80026773)
GIGA Yoshikazu  Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70144110)
ISHII Katsuyuki  Kobe University of Mercantile Marine, Faculty of Mercantile Marine, Associate Professor, 商船学部, 助教授 (40232227)
KOIKE Shigeaki  Saitama University, Faculty of Science, Professor, 理学部, 教授 (90205295)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥15,500,000 (Direct Cost: ¥15,500,000)
Fiscal Year 2002: ¥4,800,000 (Direct Cost: ¥4,800,000)
Fiscal Year 2001: ¥4,700,000 (Direct Cost: ¥4,700,000)
Fiscal Year 2000: ¥6,000,000 (Direct Cost: ¥6,000,000)
KeywordsViscosity solutions / Optimal control / Curvature flow / Hamilton-Jacobi equations / Gauss curvature flow / Stochastic optimal control / Level set approach / Uniformly elliptic equations / 凸化ガウス曲率流 / 離散確率近似 / 緩和 / 積分微分方程式 / 特異摂動 / 適正粘性解 / 半連続粘性解 / 退化楕円形方程式 / 退化楕円型方程式
Research Abstract

The results obtained in our project are : (1) In 1974, W. Firey proposed the Gauss curvature flow as a mathematical model of the wearing process of a stone rolling on the beach by wave. This model assumes that the stone has a convex shape. In our research we considered the case when a stone is not convex. We introduced the convexified Gauss curvature flow which models the wearing process of a nonconvex stone rolling on the beach and established the level set approach based on viscosity solutions method. (2) We introduced weak solutions to the integral equation which describes the convexified Gauss curvature flow, proved that the uniqueness of the weak solution for the Cauchy problem, and proved its existence by a discrete stochastic approximation. (3) We studied a general stochastic optimal control problem with state constraint and proved, under relatively weak assumptions, the Lipschitz continuity and Holder continuity of the associated value function, that the value function satisfy the corresponding Bellman equation in the viscosity sense and that the state constraint problem for the Bellman equation has a unique viscosity solution. (4) We introduced the notion of proper viscosity solutions of a wide class of first-order partial differential equations including the Burgers equation, proved the unique existence of proper viscosity solutions for the class of equations, which may not have divergence form, and established the convergence of the approximation by the vanishing viscosity method to proper viscosity solutions in the sense of convergence of graphs with respect to the Hausdorff distance.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (33 results)

All Other

All Publications (33 results)

  • [Publications] Hitoshi Ishii: "Asymptotic analysis for a class of infinite systems of first-order PDE : nonlinear parabolic PDE in the singular limit"Comm.Partial Differential Equations. 28(印刷中). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hitoshi Ishii: "A class of stochastic optimal control problems with state constraint"Indiana Univ.Math.J.. 51. 1167-1197 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Toshio Mikami: "A two-dimensional random crystalline algorithm for Gauss curvature flow"Adv.In Appl.Probab.. 34. 491-504 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yoshikazu Giga: "Viscosity solutions with shocks"Comm.Pure Appl.Math.. 55. 431-480 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hitoshi Ishii: "A mathematical model of the wearing process of a nonconvex stone"SIAM J.Math.Anal.. 33. 860-876 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hitoshi Ishii: "On the rate of convergence in homogenization of Hamilton-Jacobi equations"Indiana Univ.Math.J.. 50. 1113-1129 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hitoshi Ishii and Kazufumi Shimano: "Asymptotic analysis for a class of infinite systems of first-order PDE : nonlinear parabolic PDE in the singular limit"Comm. Partial Differential Equations. 28-1/2. (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hitoshi Ishii and Paola Loreti: "A class of stochastic optimal control problems with state constraint"Indiana Univ. Math. J.. 51-5. 1167-1196 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hitoshi Ishii and Toshio Mikami: "A two-dimensional random crystalline algorithm for Gauss curvature flow"Adv. In Appl. Probab.. 34-3. 491-504 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yoshikazu Giga: "Viscosity solutions with shocks"Comm. Pure Appl. Math.. 55-4. 431-480 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Shigeaki Koike and Toshimi Takahashi: "Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients"Adv. Differential Equations. 7-4. 493-512 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hitoshi Ishii and Toshio Mikami: "A mathematical model of the wearing process of a nonconvex stone"SIAM J. Math. Anal.. 33-4. 860-876 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Italo Capuzzo Dolcetta and Hitoshi Ishii: "On the rate of convergence in homogenization of Hamilton-Jacobi equations"Indiana Univ. Math. J.. 50-3. 1113-1129 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hitoshi Ishii and Katsuyuki Ishii: "An approximation scheme for motion by mean curvature with right-angle boundary condition"SIAM J. Math. Anal.. 33-2. 369-389 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Mi-Ho Giga and Yoshikazu Giga: "Generalized motion by nonlocal curvature in the plane"Arch. Rat. Mech. Anal.. 159-4. 295-333 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Ishii, P.Loreti: "Relaxation in an L^∞ optimization problem"Proc. Royal Soc. Edinburgh. 133(未定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ishii, T.Mikami: "A level set approach to the wearing process of a nonconvex stone"Calc. Var. Partial Differential Equations. (未定).

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ishii, K.Shimano: "Asymptotic analysis for a class of infinite systems of first-order PDE : nonlinear parabolic PDE in the singular limit"Comm. Partial Differential Equations. 28・1/2(未定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ishii, P.Loreti: "A class of stochastic optimal control problems with state constraint"Indiana Univ. Math. J.. 51・5. 1167-1196 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ishii, T.Mikami: "A two-dimensional random crystalline algorithm for Gauss curvature flow"Adv. in Appl. Probab.. 34・3. 491-504 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Giga: "Viscosity solutions with shocks"Comm. Pure Appl. Math.. 55・4. 431-480 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 石井 仁司: "A mathematical model of the wearing process of a nonconvex stone"SIAM J. Math. Anal. 33・4. 860-876 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 石井 仁司: "On the rate of convergence in homogenization of Hamilton-Jacobi equations"Indiana Univ. Math. J. 50・3. 1113-1129 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 石井 仁司: "Hamilton-Jacobi equations with partial gradient and application to homoge-nization"Comm. Partial Differential Equations. 26・5-6. 983-1002 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 石井 仁司: "An a approximation scheme for motion by mean curvature with right-angle boundary condition"SIAM J. Math. Anal. 33・2. 369-389 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 儀我 美一: "Generalized motion by nonlocal curvature in the plane"Arch. Rat. Mech. Anal. 159・4. 295-333 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 小池 茂昭: "Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients"Adv. Differential Equations. 7・4. 493-512 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 石井仁司: "A PDE approach to stochastic invariance"Discrete Contin.Dynam.Systems. 6・1. 651-664 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 石井仁司: "A Dirichlet type problem for nonlinear degenerate elliptic equations arising in time-optimal stochastic control"Adv.Math.Sci.Appl.. 10・1. 329-352 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 石井仁司: "A characterization of the existence of solutions for Hamilton-Jacobi equations in ergodic control problems with applications"Appl.Math.Optim.. 42・1. 35-50 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 儀我美一: "Crystalline and level set flow-convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane"Free boundary problems : theory and applications, I (Chiba). 64-79 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 望月清: "Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values"Nonlinear Anal.Ser.A : Theory Methods. 39・1. 33-45 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 小池茂昭: "Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem"SIAM J.Control Optim.. 38・2. 470-481 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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