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2001 Fiscal Year Final Research Report Summary

Morse theory of variational problems

Research Project

Project/Area Number 11440059
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKeio University

Principal Investigator

KIKUCHI Norio  Keio Univ. Fac. of Sci. and Tech., Professor, 理工学部, 教授 (80090041)

Co-Investigator(Kenkyū-buntansha) MISAWA Masashi  Kumamoto Univ., Fac. of Sci., Associate Professor, 理学部, 助教授 (40242672)
SHIMOMURA Shun  Keio Univ., Fac. of Sci. and Tech., Professor, 理工学部, 教授 (00154328)
TANI Atushi  Keio Univ., Fac. of Sci. and Tech., Professor, 理工学部, 教授 (90118969)
星野 慶介  千葉工業大学, 工学部, 講師
OMATA Seiro  Kanazawa Univ., Fac. of Sci., Associate Professor, 理学部, 助教授 (20214223)
ISHIKAWA Shirou  Keio Univ., Fac. of Sci. and Tech., Associate Professor (10051913)
Project Period (FY) 1999 – 2001
Keywordsdiscrete Morse flow / Morse(variational)flow / difference partial differential equation of elliptic-parabolic type / Rothe's approximation / local estimate / De Giorgi-Nash estimate / Campanato estimate / Gehring-Giaquinta-Modica higher integrability
Research Abstract

For the construction of Morse (variational) flows, we introduced the Discrete Morse Flows method, which has been adopted and named Minimizing Movements method by De Giorgi. This approach has some merits in construction of Morse flows, since we can make the most of minimizers to a series of variational functionals inductively introduced starting with initial data. By using this method, we construct Morse flows to variational problems of harmonic map type, in the treatment of which we have noticed the method is available under the weaker assumptions on initial and boundary data. We are trying to approach the construction problems of Morse flows to harmonic map variational problems between metric manifolds, which have no smoothness of the coefficients of the second order differential operators. In the analysis of Discrete Morse Flows, we carry out local estimates of time-discrete partial differential equations of elliptic-parabolic type, so that we have achieved De Giorgi-Nash type Holder estimates and Campanato estimates, and the higher integrability to the gradients of Gehring type. Such kinds of estimates are proved to hold independently of the approximation schemes, which enables to construct Morse flows through Discrete Morse Flows. In conclusion of a series of these research work, we have recognized the characteristics of Discrete Morse Flows method that it can be made use of to the construction problems of Morse flows, to which Leray-Schauder with Schauder estimates cannot be directly applied.

  • Research Products

    (26 results)

All Other

All Publications (26 results)

  • [Publications] N.Kikuchi: "Convergence of Rothe's method in Holder spaces"Applications of Mathematics. (to appear). (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] N.Kikuchi: "Holder estimates of solutions for difference differential equations of elliptic-parabolic type"The Journal of Geometric Analysis. 10-3. 525-538 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] N.Kikuchi: "On the higher integrability of the gradients of the solutions to difference partial differential systems of elliptic-parabolic type"Z. angew. Math. Phys.. 51. 290-303 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Nagasawa, K.Nakane, S.Omata: "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
  • [Publications] M.Misawa: "Local Holder regularity of gradients for evolutional p-Laplacian systems"Annali di Matematica pura ed applicata. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Misawa: "Partial regularity results for evolutional p-Laplacian systems with natural growth"manuscripta mathematica. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] N.Kikuchi: "Campanato estimates of the solutions to difference divergence formed partial differential systems ofelliptic-parabolic type"Ann. Sc. Norm. Sup. Pisa.. To appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] N.Kikuchi: "Construction of harmonic map flows through the method of ditecrete Morse flows"Proceedings of Algoritimy. (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] N.Kikuchi: "Convergence of Rothe's method in Holder spaces"Applications of Mathematics. To appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] N.Kikuchi: "Holder estimates of solutions for difference differential equations of elliptic-parabolic type"The Journal of Geometric Analysis. 10-3. 525-538 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] J.Haga and N.Kikuchi: "On the higher integrability of the gradients of the solutions to difference partial differential systems of elliptic-parabolic type"Z. angew. Math. Phys.. 51. 290-303 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Kinami, M.Mei and S.Omata: "Asymptotic Toward Diffusion Waves of the Solutions for Benjamin-Bona-Mahony-Burgers Equations"Applicable Analysis. 75, (3-4). 317-340 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Imai, K.Kikuchi, K.Nakane, S.Omata and T.Tachikawa: "A Numerical Approach to the Asymptotic Behavior of Solutions of a One-Dimensional Free Boundary Problem of Hyperbolic Type"Japan Journal of Industrial and Applied Mathematics. 18, No.1. 43-58 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Omata, H.Iwasaki and K.Kawagoe: "Numerical calculations for the eikonal equation via the discrete Morse semiflow with Ginzburg-Landau energy"Adv. Math. Sci. Appl.. 11, No.2. 781-790 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Omata and S.Kinami: "A numerical approach to the eikonal equation"Nonlinear Analysis. 47. 3795-3802 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Nagasawa, K.Nakane and S.Omata: "Numerical Computations for motion of vortices governed by a Hyperbolic Ginzburg-Landau System"Nonlinear Analysis. 51, No.1. 67-77 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Misawa: "Local Holder regularity of gradients for evolutional p-Laplacian systems"Annali di Matematica pura ed applicata. To appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Misawa: "Partial regularity results for evolutional p-Laplacian systems with natural growth"Manuscripta mathematica. To appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Ito and A.Tani: "The initial value problem for the non-homogeneous Navier-Stokes equations with general slip boundary condition"Proceedings of the Royal Society of Edinburgh. 130A. 827-835 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Ogawa and A.Tani: "Free Boundary Problem for an Incompressible Ideal Fluid with Surface Tension"Mathematical Models & Methods in Applied Sciences. Vol.12, No.12. 1725-1740 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Ito and A.Tani: "A Boundary Value Problem for an Infinite Elastic Strip with a Semi-infinite Crack"Journal of Elasticity. 66. 193-206 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Kusaka and A.Tani: "Classical Solvability of the Two-phase Stefan Problem in a Viscous Incompressible Fluid Flow"Mathematical Models and Methods in Applied Sciences. Vol.12, No.3. 365-391 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Kurokiba, N.Tanaka and A.Tani: "Existence of solution for Eguchi-Oki-Matsumura equation describing phase separation and order-disorder trasition in binary alloys"J. Math. Anal. Appl.. 272. 448-457 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Shimomura: "Painleve Property of a Degenerate Gamier System of (9/2)-type and of a Certain Fourth Order Non-Linear Ordinary Differential Equation"Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). XXIX. 1-17 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Shimomura,: "On Deficiencies of Small Functions for Painleve Transcendents of the Fourth Kind"Ann. Acade. Sci. Fennica Math.. 27. 109-120 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Ishikawa: "A dynamical system theoretical approach to equilibriium statistical mechanics"Far East Journal of Dynamical Systems. 3. 9-28 (2001)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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