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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
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,700,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥300,000)
Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordsdegenerate elliptic equation / singular solution / P-harmonic / weighted Sobolev ineauality / linearization / Hardy ineouality / Rellich inequality / missing terms / 最良定数 / 変分問題 / ハーディの不等式 / 非線形楕円型方程式 / P-調和作用素 / ハーディーの不等式 / 最小解
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.

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (27 results)

All 2008 2007 2006 2005 Other

All Journal Article (21 results) (of which Peer Reviewed: 4 results) Presentation (4 results) Book (1 results) Remarks (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
    • Related Report
      2007 Final Research Report Summary
    • 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
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Kato's inequalities for degenerate quasilinear elliptic operators2008

    • Author(s)
      堀内利郎
    • Journal Title

      Kyungpook Mathematical Journal 48-1

      Pages: 15-24

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [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
    • Related Report
      2007 Final Research Report Summary
  • [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
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Lattice-free finite difference method for numerical solution of inverse heat conduction problem.2007

    • Author(s)
      Iijima, K.
    • Journal Title

      Inverse Probl. Sci. Eng. 15-2

      Pages: 93-106

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Missing terms in Classical Inequalities.2007

    • Author(s)
      堀内利郎
    • Journal Title

      数理解析研究所 講究録 1553

      Pages: 1-16

    • Related Report
      2007 Annual Research Report
  • [Journal Article] Kato's Inequalities for Degenerate Quasilinear Elliptic Operators2007

    • Author(s)
      Horiuchi, Toshio
    • Journal Title

      Kyungpook Mathematical Journal (未定)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Lattice-free finite difference method for numerical solution of inverse heat conduction problem2007

    • Author(s)
      Iijima, Kentaro
    • Journal Title

      Inverse Probl. Sci. Eng. 15

      Pages: 93-106

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Erratum to "On the minimal solution for quasilinear degenerate elliptic equation and its blow-up2006

    • Author(s)
      Horiuchi, Toshio
    • Journal Title

      Journal of Mathematics of Kyoto University 46-1

      Pages: 231-234

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Lp-boundedness of Bergman projections for α-parabolic operators2006

    • Author(s)
      Nishio, Masaharu
    • Journal Title

      Advanced Studies in Pure Mathematics 44

      Pages: 305-318

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Construction of an atomic decomposition for functions with compact support2006

    • Author(s)
      Nakai, Eiichi
    • Journal Title

      J. Math. Anal. Appl. 313

      Pages: 730-737

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The Campanato, Morrey and Holder spaces on spaces of homogeneous type2006

    • Author(s)
      Nakai, Eiichi
    • Journal Title

      Studia Math. 176

      Pages: 1-19

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Erratum to "On the minimal solution for quasilinear degenerate elliptic equation and its blow-up2006

    • Author(s)
      堀内利郎
    • Journal Title

      Journal of Mathematics of Kyoto University (掲載決定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Lp-boundedness of Bergman projections for α-parabolic operators2006

    • Author(s)
      Nishio, Masaharu
    • Journal Title

      Advanced Studies in Pure Mathematics (掲載決定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Construction of an atomic decomposition for functions with compact support2006

    • Author(s)
      Nakai, Eiichi
    • Journal Title

      J.Math.Anal.Appl. 313

      Pages: 730-737

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Sharp Remainder terms of Hardy-Sobolev inequalities2005

    • Author(s)
      安藤 広
    • Journal Title

      Math. J. Ibaraki Univ. 37

      Pages: 39-52

    • NAID

      130000886481

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • 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
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Sharp Remainder terms of Hardy-Sobolev inequalities2005

    • Author(s)
      安藤広
    • Journal Title

      Mathematical Journal of Ibaraki University 37

      Pages: 39-52

    • NAID

      130000886481

    • Related Report
      2005 Annual Research Report
  • [Journal Article] α-Bergman spaces2005

    • Author(s)
      Nishio, Masaharu
    • Journal Title

      Osaka Journal of Mathematics 42

      Pages: 133-162

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Structure of the solution set for a vector valued elliptic boundary value problem with a Lipschitz nonlinear term2005

    • Author(s)
      Sato, Tokushi
    • Journal Title

      Nonlinear Analysis 61

      Pages: 551-557

    • Related Report
      2005 Annual Research Report
  • [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
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [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
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [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
    • Related Report
      2007 Final Research Report Summary
  • [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
    • Related Report
      2007 Final Research Report Summary
  • [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
    • Related Report
      2007 Final Research Report Summary
  • [Remarks] 茨城大学研究者情報総覧

    • URL

      http://info.ibaraki.ac.jp/scripts/websearch/index.htm

    • Related Report
      2007 Annual Research Report

URL: 

Published: 2005-04-01   Modified: 2016-04-21  

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