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Theory and applications of viscosity solutions

Research Project

Project/Area Number 09440067
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionTOKYO METROPOLITAN UNIVERSITY

Principal Investigator

ISHII Hitoshi  Tokyo Metropolitan University, Science, Professor, 理学研究科, 教授 (70102887)

Co-Investigator(Kenkyū-buntansha) TOMITA Yoshihito  Kobe University of Mercantile Marine, Mercantile Marine Science, Professor, 商船学部, 教授 (50031456)
GIGA Yoshikazu  Hokkaido University, Science, Professor, 大学院・理学研究科, 教授 (70144110)
MOCHIZUKI Kiyoshi  Tokyo Metropolitan University, Science, Professor, 大学院・理学研究科, 教授 (80026773)
ISHII Katsuyuki  Kobe University of Mercantile Marine, Mercantile Marine Science, Associate Professor, 商船学部, 助教授 (40232227)
KOIKE Shigeaki  Saitama University, Science, Associate Professor, 理学部, 助教授 (90205295)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥13,400,000 (Direct Cost: ¥13,400,000)
Fiscal Year 1999: ¥4,200,000 (Direct Cost: ¥4,200,000)
Fiscal Year 1998: ¥4,300,000 (Direct Cost: ¥4,300,000)
Fiscal Year 1997: ¥4,900,000 (Direct Cost: ¥4,900,000)
Keywordsviscosity solutions / Hamilton-Jacobi equations / degenerate elliptic equations / curvature flow / optimal control / stochastic optimal control / homogenization / semicontinuous viscosity solutions / ガウス曲率流 / 確率微分方程式 / 不変領域 / 摩耗のモデル / Lax-Hopfの公式 / 非線形偏微分方程式 / ハミルトン・セコビ方程式 / ダイナミック・プログラミング
Research Abstract

The results obtained are summarized as follows. 1. We introduced a method of constructing an approximate feedback control for state-constraint control problems via viscosity solutions of the corresponding Hamilton-Jacobi equations. 2. The uniqueness and existence theorem due to Barron-Jensen on semicontinuous viscosity solutions of Hamilton-Jacobi equations is a fundamental tool in characterizing value functions in optimal control when the value functions are semicontinuous. We established a theorem similar to the Barron-Jensen theorem in Hilbert spaces. 3. We considered the Hamilton-Jacobi equation in ergodic control and gave a characterization of the existence of viscosity solutions of the Hamilton-Jacobi equation through a kind of value function of the corresponding ergodic optimal control. 4. In the Barron-Jensen theory of semicontinuous viscosity solutions the convexity of Hamiltonians is a key assumption. We introduced a notion of semicontinuous viscosity solution for Hamilton-Jacobi equations with non-convex Hamiltonian for which nice uniqueness and existence properties hold. 5. We studied the solvability, uniqueness, smoothness of solutions of Bellman equations in risk-sensitive stochastic control as well as the relation between its singular limit and a differential game. 6. We introduced a geometric approximation scheme for Gauss curvature flow of a convex body and proved its convergence. 7. We proved the equivalence between the invariance of a controlled stochastic differential equation with respect to a compact set and the restriction property to the compact set of viscosity solutions of the corresponding Bellman equation. 8. We studied the waiting time phenomena for Gauss curvature flow of a convex set and proved that if two principal curvatures vanish at a point on the initial surface then the waiting time of the point is positive.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (33 results)

All Other

All Publications (33 results)

  • [Publications] 石井仁司: "On ε-optimal controls for state constraint problems"Ann. Inst. H. Poincare. (未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 石井仁司: "Hopf-Lax formulas for semicontinuous data"Indiana Univ. Mayth. J.. 48・3. 994-1035 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 石井仁司: "Waiting time effects for Gauss curvature flows"Indiana Univ. Math. J.. 48・1. 311-334 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 望月清: "Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values"Nonlinear Analysis. 39. 33-45 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 冨田義人: "Unbounded viscosity solutions of nonlinear second order PDE's"Advances in Math. Sciences Appl.. 10・2(未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 儀我美一: "On strong maximum principle and large time behavior of generalized mean curvature flow with the Neumann boundary condition"J. Differential Equations. 154. 107-131 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Ishii, S. Koike: "On ε-optimal controls for state constraint problems"Ann. Inst. H. Poincare. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] O. Alvarez, E. N. Barron, H. Ishii: "Hopf-Lax formulas for semicontinuous data"Indiana Univ. Math. J.. 48. 994-1035 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] D. Chopp, L. C. Evans, H. Ishii: "Waiting time effects for Gauss curvature flows"Indiana Univ. Math. J.. 48. 311-334 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Ishii, G. E. Pires, P. E. Souganidis: "Threshold dynamics type approximation schemes for propagating fronts"J. Math. Soc. Japan. 51. 267-308 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Horie, H. Ishii: "Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure"Indiana Univ. Math. J.. 47. 1011-1058 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Mochizuki, K. Mukai, Q. Huang: "Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values"Nonlinear Analysis. 39. 33-45 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Ishii, Y. Tomita: "Unbounded viscosity solutions of nonlinear second order PDE's"Advances in Math. Sci. Appl.. 10-2,(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. -H. Giga, Y. Giga: "Stability for evolving graphs by nonlocal weighted curvature"Comm. PDEs. 24. 109-184 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Giga, M. Ohnuma, M. -H. Sato: "On strong maximum principle and large time behavior of generalized mean curvature flow with the Neumann boundary condition"J. Differential Equations. 154. 107-131 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 石井 仁司: "On ε-optimal controls for state constraint problems"Ann. Inst. H. Poincare. (未定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 石井 仁司: "Hopf-Lax formulas for semicontinuous data"Indiana Univ. Math. J.. 48・3. 994-1035 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 石井 仁司: "Waiting time effects for Gauss curvature flows"Indiana Univ. Math. J.. 48・1. 311-334 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 望月 清: "Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values"Nonlinear Analysis.. 39. 33-45 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 冨田 義人: "Unbounded viscosity solutions of nonlinear second order PDE's,"Advances in Math Sciences Appl.. 10・2(未定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 儀我 美一: "On strong maximum principle and large time behaviour of generalized mean cuvature flow with the Neumann boundary condition"J. Differential Equations. 154. 107-131 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 石井仁司: "Homogenization of the Cauchy problem for Hamilton-Jacobi equations" Stochastic Analysis,Control,Optimization and Applications:a volume in honor of W.H.Fleming. 305-324 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 石井仁司: "Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure" Indiana Univ.Math.J.47. 1011-1058 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 石井仁司: "Threshold dynamics type approximations schemes for propagating fronts" J.Math.Soc.Japan. 印刷中.

    • Related Report
      1998 Annual Research Report
  • [Publications] 望月 清: "Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations." Methods Appl.Anal.5・2. 109-124 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 儀我美一: "Evolving graphs by singular weighted curvature" Arch.Rational Mech.Anal.141・2. 117-198 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 石井克幸: "Regularity and convergence of crystalline motion" SIAM J. Math. Anal.30・1. 19-37

    • Related Report
      1998 Annual Research Report
  • [Publications] 石井 仁司: "Some properties of ergodic attractors for controlled dynamical systems" Discrete Contin.Dyram.Systems. 4.1. 43-54 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] P.-L.Lions: "Un contre-exemple pour la theorie du controle ergodique en dimension infinie" C.R.Acad.Sci.Paris Ser.I Math.325・1. 37-41 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 石井 仁司: "Threshold dynamics type approximation schemes for propagating fronts" J.Math.Soc.Japan. (未定).

    • Related Report
      1997 Annual Research Report
  • [Publications] 石井 仁司: "An approximation scheme for Gauss curvature flow" 数理解析研究所講究録. (未定).

    • Related Report
      1997 Annual Research Report
  • [Publications] 望月 清: "Critical exponent and critical blow-up for quasilinear parabolic equations" Israel J.Math.98. 141-156 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 儀我 美一: "On instant extinction for very fast diffusion equations" Discrete Contin.Dynam.Systems.3・2. 243-250 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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