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2007 Fiscal Year Final Research Report Summary

On the linearization of quasilinear degenerate elliptic equations and the structure of singular solutions

Research Project

Project/Area Number 17540146
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

HORIUCHI Toshio  Ibaraki University, College of Science, Professor (80157057)

Co-Investigator(Kenkyū-buntansha) ONISHI Kazuei  Ibaraki University, College of Science, Professor (20078554)
SHIMOMURA Katunori  Ibaraki University, College of Science, Assistant Professor (00201559)
ANDO Hiroshi  Ibaraki University, College of Science, Lecturer (60292471)
NAKAI Eiichi  Osaka Education University, Fac of Education, Professor (60259900)
SATO Tokushi  Tohoku University Graduate School, Institute of Scaence, Assistant Professor (00261545)
Project Period (FY) 2005 – 2007
Keywordsdegenerate elliptic equation / singular solution / P-harmonic / weighted Sobolev ineauality / linearization / Hardy ineouality / Rellich inequality / missing terms
Research Abstract

1. On the structure of singular solutions for degenerate elliptic equations:
(1) We studied various types of Hardy-Sobolev-Rellich inequalities, and improved them by finding out sharp missing terms. We applied them to the study of Blow-up solutions.
(2) We studied the weighted Hardy-Sobolev inequalities. For the Caffarelli-Kohn-Nirenberg type weight functions, the inequalities have been improved by adding sharp remainder terms. In particular for the critical weight case and the supercritical case new CKN-inequalities were obtained.
(3) We studied CKN-inequalities and extended them to the critical and supercritical case. This gives not only new imbedding theorems but also new framework for the PDE problems. We also established Existence and Nonexistence results on the extremals, asymptotic behaviors of the best constant and some qualitative properties of extremals. We applied our inequalities to study the boundary value problems having singular potential with the best constant as its coefficient in nontrivial functional framework.
2. On the linearization of quasi-linear elliptic PDE.
P-harmonic equation with strong positive nonlinear terms in the right-hand side has been studied systematically and established the unique existence results of minimal solutions. Moreover we studied very well linearized operators at the minimal solutions to made clear the coercivity and positivity of the operator and to construct the theory of Bifurcation.

  • Research Products

    (11 results)

All 2008 2007 2005

All Journal Article (6 results) (of which Peer Reviewed: 2 results) Presentation (4 results) Book (1 results)

  • [Journal Article] Kato's inequalities for degenerate quasilinear elliptic operators2008

    • Author(s)
      堀内利郎
    • Journal Title

      Kyungpook Mathcmatical Journal 48-1

      Pages: 15-24

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Kato''s inequalities for degenerate quasilinear elliptic Operators2008

    • Author(s)
      T. Horiuchi
    • Journal Title

      Kyungpook Mathematical Journal 48-1

      Pages: 15-24

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Missing terms in Classical Inequalites.2007

    • Author(s)
      堀内利郎
    • Journal Title

      数理解析研究所 講究録 1553

      Pages: 1-16

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Missing terms in Classical Inequalities2007

    • Author(s)
      T. Horiuchi
    • Journal Title

      Kyoto LTniv ersity. R. I. M. S. Koukyu_roku 1553

      Pages: 1-16

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Sharp Remainder terms of Hardy-Sobolev inequalities2005

    • Author(s)
      安藤 広
    • Journal Title

      Math. J. Ibaraki Univ. 37

      Pages: 39-52

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Sharp Remainder terms of Hardy-Soholev inequalities2005

    • Author(s)
      H. Ando
    • Journal Title

      Math. J. Ibaraki Univ 37

      Pages: 39-52

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 重み付きハーディー・ソボレフ・レリッヒの不等式について2007

    • Author(s)
      安藤 広
    • Organizer
      日本数学会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] 重み付きハーディー・ソボレフの不等式の精密化について2007

    • Author(s)
      安藤 広
    • Organizer
      日本数学会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] On the weighted Hardy-Sobolev-Rellich inequalities2007

    • Author(s)
      H. Ando
    • Organizer
      Meeting for Mathematical Society of Japan
    • Place of Presentation
      Tohoku Univ
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] On the weighted Hardy-Sobolev inequalities2007

    • Author(s)
      H. Ando
    • Organizer
      Meeting for Mathematical Society of Japan
    • Place of Presentation
      Tohoku Univ
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] 関数解析の基礎2005

    • Author(s)
      堀内利郎
    • Total Pages
      296
    • Publisher
      内田老鶴圃
    • Description
      「研究成果報告書概要(和文)」より

URL: 

Published: 2010-02-04  

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