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2007 Fiscal Year Final Research Report Summary

studies on the relationship of the structure of manifolds and p-harmonic functions

Research Project

Project/Area Number 16540208
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TAKEUCHI Hiroshi  Shikoku University, Faculty of Management and Information, Professor (20197271)

Co-Investigator(Kenkyū-buntansha) SAKAI Takashi  Okayama University of Science, Faculty of Science, Professor (70005809)
KATSUDA Atsushi  Okayama University, Graduate School of Natural Science and Technology, Associate Professor (60183779)
Project Period (FY) 2004 – 2007
Keywordsp-harmonic function / p-harmonic map / Riemannian manifold / graph
Research Abstract

P-Laplacian Δ_p(1<P<∞)is defined as operator acting on functions on Riemannian manifolds. The p-harmonic function u is defind by Δ_<p>u=div(|∇u|^<p-2> ∇u)=0. In the case of p=2, it becomes the usual harmonic map. We may consider the p-harmonic function the extension of harmonic function. In fact, the p-harmonic function is a critical point of the p-energy functional. Euler-Lagrange equation of it is that of the p-harmonic function. Because this equation is a nonlinear elliptic partial equation, it is hard to handle. We can define the p-Laplacian on graphs also, and define the p-harmonic function on graphs.
We consider the spectrum of the p-Laplacian on graphs, p-harmonic morphisms between two graphs, and estimates for the solutions of p-Laplace equations on graphs. More precisely we prove a Ceeger type inequality and a Brooks type inequality for infinite graphs. We showed p-harmonic morphisms and horizontal conformal maps between two graphs are equivalent. We give some estimates for solutions of p-Laplace equations, which coincide with Green kernels in the case of p=2.
Harmonic maps flow and p-harmonics flow are closely related to harmonic maps and p-harmonic maps.
The stationary state of harmonic maps flow becomes the harmonic map, and the stationary state of p-harmonic maps flow becomes the p-harmonic map. But they do not necessarily converge, but blow-up of the solutions happen. We report this phenomena as research notes in Bulletin of Shikoku University.

  • Research Products

    (7 results)

All 2008 2007 2004 2003

All Journal Article (7 results) (of which Peer Reviewed: 2 results)

  • [Journal Article] 調和写像流について2008

    • Author(s)
      竹内 博
    • Journal Title

      四国大学紀要, 自然科学編 26

      Pages: 13-17

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Notes on the harmonic map heat flows(In Japanese)2008

    • Author(s)
      Hiroshi Takeuchi
    • Journal Title

      Bulletin of Shikoku University Ser. B No 26

      Pages: 13-17

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Cut loci and distance functions2007

    • Author(s)
      J.Itoh, T.Sakai
    • Journal Title

      Math.J.Okayama Univ. 49

      Pages: 65-92

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Cut loci and distance functions2007

    • Author(s)
      J. Itoh, T. Sakai
    • Journal Title

      Math. J. Okayama Univ. 49

      Pages: 65-92

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Boundary regularity for Ricci equation,geometric convergence,and Gel'fand's inverse boundary problem2004

    • Author(s)
      M.Anderson, A.Katsuda, et. al.
    • Journal Title

      Invent.Math. 158

      Pages: 261-321

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Boundary regularity for Ricci equation, geometori Convergence, and Gel'fand's inverse boundary problem2004

    • Author(s)
      M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas, M. Taylor
    • Journal Title

      Invent. Math. 158

      Pages: 261-321

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The spectrum of the p-Laplacian and p-harmonic morphisms on graphs2003

    • Author(s)
      Hiroshi Takeuchi
    • Journal Title

      Illinois J. Math. Vol.47, No.3

      Pages: 939-955

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

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Published: 2010-02-04  

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