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Study of viscosity solutions for partial differential equations with subdifferential

Research Project

Project/Area Number 13640200
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TANAKA Naoto (2003)  Fukuoka University, Faculty of Science, Assistant Professor, 理学部, 助教授 (00247222)

山田 直記 (2001-2002)  福岡大学, 理学部, 教授 (50030789)

Co-Investigator(Kenkyū-buntansha) MARUO Kenji  Kobe University, Faculty of Maritime Science, Professor, 海事科学部, 教授 (90028225)
KUROKIBA Masaki  Fukuoka University, Faculty of Science, Assistant, 理学部, 助手 (60291837)
YAMADA Naoki  Fukuoka University, Faculty of Science, Professor, 理学部, 教授 (50030789)
ISHII Katsuyuki  Kobe University, Faculty of Maritime Science, Assistant Professor, 海事科学部, 助教授 (40232227)
田中 尚人  福岡大学, 理学部, 助教授 (00247222)
草野 尚  福岡大学, 理学部, 教授 (70033868)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordsviscosity solution / subdifferential
Research Abstract

The aim of present research is to extend the notion of viscosity solutions to multi-valued nonlinear partial differential equations (PDE's) with subdifferential and to investigate the existence, uniqueness and other properties of solutions. The theory of viscosity solutions has been applied to wide class of frilly nonlinear PDE's. Among them, the obstacle problem is an important class of applications of viscosity solutions. We have put emphasis of the research on following three points:
1. We formulate various PDE's by subdifferential and apply the theory of viscosity solutions for it.
2. We compare the formulation by viscosity solution with known ones and study an advantage for it.
3. We prove solvability of various nonlinear PDE's and discuss the relation between the methods by viscosity solution.
It is shown that the notion of viscosity solutions are extended for nonlinear second order PDE's with subdifferential whose typical example is obstacle problem, and unique, existence theorem and stability of the solution are proved. Moreover, we investigate convergence of Yoshida approximation for subdiffrential term.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (60 results)

All Other

All Publications (60 results)

  • [Publications] 伊藤茂治, 田中尚人, 谷温之: "The initial value problem for the Navier-Stokes equations with general bound slip boundary condition in Holder spaces"Journal of Mathematical Fluid Mechanics. 5. 276-301 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 田中尚人, 谷温之: "Surface waves for a compressible viscous fluid"Journal of Mathematical Fluid Mechanics. 5. 303-363 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 山田直記: "A comparison theorem for viscosity solutions of first order Subdifferentioal equations"Fukuoka University Science Report. 31. 1-5 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 丸尾健二, 富田義人: "Radial viscosity solutions of the Dirichlelt problem for semilinear degenerate elliptic equations"0saka Journal of Mathematics. 38. 737-757 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 丸尾健二, 富田義人: "Remarks on viscosity solutions of the Dirichlet problem for quasilinear dgenerate elliptic equations"Funkcialaj Ekvacioj. 45. 89-101 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 富田義人, 丸尾健二: "Unbounded radially symmetric viscosity solutions of semilinear degenerate elliptic equations"Science Mathematicae Japonicae. 58. 15-31 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 石井克幸, 山田直記: "Nonlinear second order elliptic pdes with subdifferential"Advances in Mathematical Sciences and Applications. 11. 477-497 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 石井克幸, 石井仁司: "An application scheme for motion by mean curvature with right-angle boundary condition"SIAM Journal of Mathematical Analysis. 33. 369-389 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 黒木場正城, 田中尚人, 谷温之: "Existence of solutions for Eguchi-Oki-Matsumrua equation describing phase separation and order-disorder transition in binary alloys"Journal of Mathematical Analysis and Applications. 272. 448-457 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 黒木場正城, 小川卓克: "Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type"Differential and Integral Equations. 16. 427-452 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Jaros, 草野尚, 吉田範人: "Picone-type inequalities for nonlinear elliptic equations and their applications"Journal of Inequalities and Applications. 6. 387-404 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Jaros, 草野尚, 吉田範人: "Oscillation properties of solutions of a class of nonlinear parabolic equations"Journal of Computations and Applied Methematics. 146. 277-284 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 草野尚, 内藤学: "On the number of zeros of nonscillatory solutions to half-linear ordinary differential equations involving a parameter"Transactions of American Mathematical Society. 354. 4751-4767 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Jaros, 草野尚: "On white hole solutions of a class of nonlinear ordinary differential equations of the second order"Funkcialaj Ekvacioj. 45. 319-339 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Jaros, 草野尚, 谷川智幸: "Nonscillation theory for second order half-linear differential equations in the framework of regular variation"Results in Mathematics. 43. 129-149 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Itoh, N.Tanaka, A.Tani.: "The initial value problem for the Navier-Stokes equations with general slip boundary condition in Holder spaces."Journal of Mathematical Fluid Mechanics. 5. 276-301 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Tanaka, A.Tani.: "Surface waves for a compressible viscous fluid."Journal of Mathematical Fluid Mechanics. 5. 303-363 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Yamada: "A comparison theorem for viscosity solutions of first order Subdifferential equations."Fukuoka University Science Report. 31. 1-5 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Maruo, Y.Tomita.: "Radial viscosity solutions of the Dirichlet problem for semilinear degenerate elliptic equations"Osaka Journal of Mathematics. 38. 737-757 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Maruo, Y.Tomita.: "Remarks on viscosity solutions of the Dirichlet problem for quasilinear degenerate elliptic equations"Funkcialaj Ekvacioj. 45. 89-101 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Maruo, Y.Tomita.: "Unbounded radially symmetric viscosity solutions of semilinear degenerate elliptic equations"Scientiae Mathematicae Japonicae. 58. 15-31 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Ishii, N.Yamada: "Nonlinear second order elliptic pdes with subdifferential"Advances in Mathematical Sciences and Applications. 11. 477-497 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Ishii, H.Ishii: "An application scheme for motion by mean curvature with right-angle boundary condition"SIAM Journal of Mathematical Analysis. 33. 369-389 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kurokiba, N.Tanaka, N.Tani.: "Existence of solutions for Eguchi-Oki-Matsumura equation describing phase separation and order-disorder transition in binary alloys"Journal of Mathematical Analysis and Applications. 272. 448-457 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kurokiba, T.Ogawa.: "Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type"Differential and Integral Equations. 16. 427-452 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano, N.Yoshida.: "Picone-type inequalities for nonlinear elliptic equations and their applications"Journal of Inequalities and Applications. 6. 387-404 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano, N.Yoshida.: "Oscillation properties of solutions of a class of nonlinear parabolic equations"Journal of Computations and Applied Mathematics. 146. 277-284 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kusano, M.Naito.: "On the number of zeros of nonoscillatory solutions to half-linear ordinary differential equations involving a parameter"Transactions of American Mathematical Society. 354. 4751-4767 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano.: "On white hole solutions of a class of nonlinear ordinary differential equations of the second order"Funkcialaj Ekvacioj. 45. 319-339 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano, T.Tanigawa.: "Nonoscillation theory for second order half linear differential equations in the framework of regular variation"Results in Mathematics. 43. 129-149 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 伊藤茂治, 田中尚人, 谷温之: "The initial value problem for the Navier-Stokes equations with general Slip boundary condition in Holder spaces"J.Math.Fluid Mech.. 5. 276-301 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 田中尚人, 谷温之: "Surface waves for a compressible viscous fluid"J.Math.Fluid Mech.. 5. 303-363 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 富田義人, 丸尾健二: "Unbounded radially symmetric viscosity solutions of semilinear degenerate elliptic equations"Sci.Math.Jpn.. 58. 15-31 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 石井克幸, 中村武: "An error estimate to Bence-Merriman-Osher algorithm for motion by mean curvature"神戸商船大学紀要、第2類、商船・理工学編. 51. 105-115 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 黒木場正城, 小川卓克: "Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type"Diff.Integral Equations. 16. 427-452 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] J.Jaros, 草野尚, 谷川智幸: "Nonoscillation theory for second order half-linear differential equations in the framework of regular variation"Result.Math.. 43. 129-149 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 山田 直記: "A comparison theorem for viscosity solutions of first order subdifferential equations"Fukuoka University Science Reports. 31(1). 1-5 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 石井 克幸, 山田 直記: "Nonlinear second order elliptic pde s with subdifferential"Advances in Mathematical Sciences and Applications. 12. 477-497 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 草野 尚, J.Jaros, 吉田範夫: "Picone-type inequakities for nonlinear elliptic equations and their applications"J. Inequal. Appl.. 6. 387-404 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 草野 尚, J.Jaros, 吉田範夫: "Oscillation properties of solutions of a class of nonlinear parabolic equations"J. Comput. Appl. Math.. 146. 277-284 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 草野 尚, 内藤 学, F.Wu: "On the oscillation of 4-dimensional Emden-Fowler differential systems"Advances in Mathematical Sciences and Applications. 11. 685-719 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 黒木場 正城: "江口タイプ方程式の解の存在について"京都大学数理解析研究所講究録. 1204. 107-113 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 黒木場 正城, 田中 尚人, 谷 温之: "Existence of solutions for Eguchi-Oki-Matsumura type equation describing phase separation and order-disorder transition in binary alloys"J. Math. Anal. Appl.. 272. 448-457 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 富田 義人, 丸尾 健二: "Radial viscosity solutions of the Dirichret problem for semilinear degenerate elliptic equations"Osaka J. Math.. 38. 737-757 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 富田 義人, 丸尾 健二: "Radial viscosity solutions of the Dirichret problem for semilinear degenerate elliptic equations"Proc. tenth Tokyo Conference on Nonlinear PDE. 45-58 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 富田 義人, 丸尾 健二: "Remarks on viscosity solutions of the Dirichret problem for quasilinear degenerate elliptic equations"Funkcialaj Ekvacioj. 45. 89-101 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 石井 克幸, 石井 仁司: "An application scheme for motion by mean curvature with right-angle boundary condition"SIAM J. Math. Anal.. 33. 369-389 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 伊藤成治, 田中尚人, 谷温之: "The Initial Value Problem for the Navier-Stokes Equations with General Slip Boundary Condition in Hoelder Spaces"J. Math. Flud. Mech.. (発表予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 田中尚人, 谷温之: "Surface Waves for a Commplessible Viscous Fluid"J. Math. Flud. Mech.. (発売予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 山田 直紀: "A comparison theorem for viscosity solutions of first order subdifferential equations"Fukuoka University Science Reports. 31(1). 1-5 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 石井 克幸, 山田 直紀: "Nonlinear second order elliptic pde's with subdifferential"Advances in Mathematical Sciences and Applications. (発売予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 草野 尚, J.jaros, 吉田範夫: "Picone-type inequakities for nonlinear elliptic equations and their applications"J. Inequal. Appl.. 6. 387-404 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 草野 尚, J.Jaros, 吉田範夫: "Oscillation properties of solutions of a class of nonlinear parabolic equations"J. Comput. Appl. Math.. (発表予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 草野 尚, 内藤 学, F.Wu: "On the oscillation of 4-dimensional Emden-Fowler differential systems"Advances in Mathematical Sciences and Applications. (発売予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 黒木場 正城: "江口タイプ方程式の解の存在について"京都大学数理解析研究所講究録. 1204. (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 黒木場 正城, 田中 尚人, 谷 温之: "Existence of solutions for Eguchi-Oki-Matsumura type equation describing phase separation and order-disorder transition in binary alloys"J. Math. Anal. Appl.. (発表予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 富田 義人, 丸尾 健二: "Radial viscosity solutions of the Dirichret problem for semilinear degenerate elliptic equations"Osaka J. Math.. 38. 737-757 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 富田 義人, 丸尾 健二: "Radial viscosity solutions of the Dirichret problem for semilinear degenerate elliptic equations"Proc. Tenth Tokyo Conference on Nonlinear PDE. 45-58 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 富田 義人, 丸尾 健二: "Remarks on viscosity solutions of the Dirichret problem for quasilinear degenerate elliptic equations"Funkcialaj Ekvacioj. 45. 89-101 (2002)

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

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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