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Nonlinear elliptic and parabolic PDEs, theories and applications

Research Project

Project/Area Number 14540148
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionTohoku University

Principal Investigator

ARISAWA Mariko  Tohoku University, GSIS, Assoc.Prof., 大学院・情報科学研究科, 助教授 (50312632)

Co-Investigator(Kenkyū-buntansha) URAKAWA Hajime  Tohoku University, GSIS, Prof., 大学院・情報科学研究科, 教授 (50022679)
GIGA Yoshikazu  Hokkaido University, Dept.of Maths., Prof., 大学院・理学研究科, 教授 (70144110)
MIKAMI Toshio  Hokkaido University, Dept.of Maths., Assoc.Prof., 大学院・理学研究科, 助教授 (70229657)
SEKINE Jyun  Osaka University, Dept of F.Enginerr., Assoc.Prof., 大学院・基礎工学研究科, 助教授 (50314399)
柳田 英二  東北大学, 大学院・理学研究科, 教授 (80174548)
石井 仁司  早稲田大学, 教育学部, 教授 (70102887)
長澤 壮之  東北大学, 大学院・理学研究科, 助教授 (70202223)
Project Period (FY) 2002 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥4,100,000 (Direct Cost: ¥4,100,000)
Fiscal Year 2003: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2002: ¥2,100,000 (Direct Cost: ¥2,100,000)
Keywordsnonlinear partial differential equations / optimal stochastic control / viscosity solutions / Hamilton-Jacobi-Bellman equation / mathematical finances / homogenizations / ergodic problem / numerical analysis / 粘性解理論 / 数理ファイナンス / ホモジェニゼイション / 多重スケールモデル / 非完備金融市場モデル / 最適制御理論 / エルゴード問題 / 偏微分方程式の数値解析 / 非線形退化楕円型方程式の解の平滑性 / 非線形熱方程式の解の安定性 / ショックの現れる非線形一階偏微分方程式の解の存在と一意性 / ハミルトンャコビベルマン方程式の数理ファイナンスへの応用 / 界面の発展方程式 / 変分法の数理生物学への応用 / 非線形偏微分方程式の数値解析 / 確率解析と非線形偏微分方程式の関わり
Research Abstract

This project was motivated to develop new methods, explore new problems and applications in the field of nonlinear elliptic and parabolic PDEs. In particular, we are interested in those problems arising in optimal controls of stochastic processes and deterministic dynamical systems.
In the first year, a treatment of the stochastic PDEs by the viscosity solutions approach was studied, and at the same time some asymptotic problem arising in mathematical finances were begun to be considered. The head investigator profited six months on leave at the university of Paris 9 for some collaborations on the above subjects (Inamori fund). Moreover, some numerical experiments are tried to those problems above with collaborators. In the second year, some problems in mathematical finances are studied : e.g. the optimal portfolio construction with transaction costs, and the option pricing problems for the stock process containing jump terms. The former leads a free boundary problem (including some asymptotic problems), and the latter leads a class of integro-partial differential equations. These problems are significant from theoretical and practical point views. The head investigator wrote a paper on the regularity of degenerate elliptic equations, concerning with stochastic control, and she also prepares a work on the nonlinear first order PDE with or without shocks, concerning with the problem of the stochastic PDE. The mathematical finances work written above is also in preparation.
In addition to the above, the head investigator have taught a course in GSIS, Tohoku University on nonlinear elliptic and parabolic PDEs, optimal control problems arising in mathematical finances, and the basic theory of numerical analysis.

Report

(3 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • Research Products

    (44 results)

All Other

All Publications (44 results)

  • [Publications] Mariko Arisawa: "Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions"Annales de Institut Henri Poincare, Analyse non-lineaire. vol.20/2. 293-332 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Mariko Arisawa: "Some regularity results for degenerate elliptic second-order partial"京都大学数理解析研究所講究録. 1323. 45-58 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hajime Urakawa: "The Dirichlet eigenvalue problem"Contempolary math.. (印刷中). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hajime Urakawa: "The Cheeger constant, the heat kernel and the Green kernel of an infinite graph"Monatshefte fur Mathematik.. (印刷中). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.-H.R.Tsai, Yoshikazu Giga, S.Osher: "A level set approach for computing discontinuous solutions of a class of Hamilton-Jacobi equations"Math.Comp.. 72. 159-181 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yoshikazu Giga, P.Rybka: "Berg's effect."dv.Math.Sci.ApApl.. 13. 523-537 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshio Mikami, H.Ishii: "A level set approach to the wearing process of a nonconvex stone."Calc.Var.Partial Differential Equations. 19. 53-93 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiji Yanagida, P.Polacik: "Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain."Disc.Cont.Dyn.Systems. 8. 209-218 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiji Yanagida: "Mini-maxiziers in reaction-diffusion systems with skew-gradient structure."J, Differential Equations. 179. 311-335 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeyuki Nagasawa: "Numerical computations for motion of vortices governed by a hyperbolic Ginzburg-Landau system."Nonlinear Analysis. 51-1. 67-77 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Jun Sekine: "Dynamic minimization of worst conditional expectation of shortfall."Mathematical Finance.. (印刷中). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Mariko Arisawa: "Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions"Annales de Institut Henri Poincare, Analyze non-lineaire. vol.20/2. 293-332 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Mariko Arisawa: "Some regularity results for degenerate elliptic second-order partial differential operators"1323. 45-58 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hajime Urakawa: "The Dirichlet eigenvalue problem"Contempolary math.. (in press). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hajime Urakawa: "The Cheeger Constant, the Heat Kernel, and the Green Kernel of an Infinite Graph"Monatshefte fur Mathematik. (in press). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.-H.R.Tsai, Yoshikazu Giga, S.Osher: "A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations"Math.Comp. 72. 159-181 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yoshikazu Giga, P.Rybka: "Berg's effect"Adv.Math.Sci.Appl.. 13. 523-537 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Toshio Mikami, H.Ishii: "A level set approach to the wearing process of a nonconvex stone"Calc.Var.Partial Differential Equations. 19. 53-93 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiji Yanagida, P.Polacik: "Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain"Disc.Cont.Dyn.Systems. 8. 209-218 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiji Yanagida: "Mini-maximizers for reaction-diffusion systems with skew-gradient structure"J, Differential Equations. 179. 311-335 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeyuki Nagasawa: "Numerical computations for motion of vortices governed by a hyperbolic Ginzburg-Landau system"Nonlinear Analysis. 51-1. 67-77 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Jun Sekine: "Dynamic minimization of worst conditional expectation of shortfall"Mathematical Finance. (in press). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Mariko Arisawa: "Some regularity results for degenerate elliptic second-order partial defferential operators"京都大学数理解析研究所講究録. 1323. 45-58 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Giga, P.Rybka: "Berg's effect"Adv.Math.Sci.Appl.. 13. 525-537 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.-H.R.Tsai, Y.Giga, S.Osher: "A level set approach for computing discontinuous solutions of a class of Hamilton-Jacobi equations"Math.Comp. 72. 159-181 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Urakawa: "Yang-Mills theory over compact symplectic manifolds"Annals Global Analysis Geometry. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Mikami, H.Ishii: "A level set approach to the wearing process of a nonconvex stone"Calc.Var.Partial Differential Equations. 19. 53-93 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] J.Sekine: "Dynamic minimization of worst conditional expectation of shortfall"Mathematical Finance. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] Mariko Arisawa: "Long time average reflection force and homogenization of oscillating Neumann boundary conditions"Annales de l Institut Henri Poincare/Analyse non linesire. vol20/2. 293-332 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Mariko Arisawa: "Some regularity results for a class of fully nonlinear degenerate elliptic second order partial differential equations"京都大学数理解析研究所講究録. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Hajime Urakawa: "The Dirichiet eigenvalue problem, the finite element method and graph theory"Contemporary Mathematics, Amer. Math. Soc., 2003. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Hajime Urakawa: "The Cheeger constant, the heat kernel and the Green kernel of an infinite graph"Monatshefte fur Mathematik, 2003. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Eiji Yanagida, P.Polacik: "Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain"Disc. Cont. Dyn. Systems. 8. 209-218 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Eiji Yanagida: "Mini-maximizers in reaction-diffusion systems with skew-gradient structure"Journal of Differential Equations. 179. 311-335 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Nagasawa, T., K.Nakane, S.Omata: "Numerical computations for motion of vortices governed by a hyperbolic Ginzburg-Landau system"Nonlinear Analysis. 51・1. 67-77 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Nagawawa, T., I.Takagi: "Bifurcating critical points of bending energy with constraints related to the shape of red blood cell"Cal. Var. Partial Differential Equations. 16・1. 36-111 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Yoshikazu Giga, Y.Thai, S.Osher: "A level set approach for computing discontinuous solutions of a class of Hamilton-Jacobi equations"Math. Comp. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Yoshikazu Giga, J. Escher, K. Ito: "On a limiting motion and self-intersections for the intermediate surface duffusion flow"J. of Evolution Equations. 2. 349-364 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hitoshi 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] Hitoshi Ishii, P.Loreti: "A class of stochastic optimal control problems with state constraint"Indiana Univ. Math. J. 51・5. 1169-1198 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Shigeaki Koike, H. Morimoto: "On variational inequalities for leavable boundary-velocity control"Applied Mathematics and Optimization. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Shigeaki Koike, A. Swiech: "Maximum principle and existence of$L^p$-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms"Nonlinear Differential Equations and Applications. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Toshio.Mikami: "A uniqueness result on Cauchy problem related to jump-type Markov processes with unbounded characteristics"Stoc. Anal. Appl. 20・2. 389-413 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Toshlo Mikami: "Optimal control for absolutely continuous stochastic processes and the mass transportation problem"Elect. Comm. in Probab.. 7. 189-203

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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