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On the improvement of classical inequalities and its applications to nonlinear degenerate elliptic equations

Research Project

Project/Area Number 20540153
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  茨城大学, 理学部, 教授 (80157057)

Co-Investigator(Kenkyū-buntansha) ONISHI Kazuei  茨城大学, 理学部, 教授 (20078554)
SHIMOMURA Katunori  茨城大学, 理学部, 教授 (00201559)
NAKAI Eiichi  茨城大学, 理学部, 教授 (60259900)
ANDO Hiroshi  茨城大学, 理学部, 講師 (60292471)
YAMAGAMI Shigeru  茨城大学, 理学部, 教授 (90175654)
Co-Investigator(Renkei-kenkyūsha) HOSHIRO Toshihiko  兵庫県立大学, 教授 (40211544)
TACHIZAWA Kazuya  北海道大学, 理学研究科, 教授 (80227090)
SATO Tokushi  東北大学, 理学研究科, 助教 (00261545)
Project Period (FY) 2008 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords重み付きソボレ不等式 / 重み付きハーディ不等式 / CKN型不等式 / ミッシング・ターム / 退化楕円型方程式 / 非線形変分問題 / 楕円型偏微分方程式 / ソボレフ不等式 / ハーディ不等式 / 退化楕円型問題 / ミッシングターム / 対称性の破れ / 非線形楕円型方程式 / 変分問題 / 退化楕円型 / バーディ不等式
Research Abstract

(1) The classical Hardy-Sobolev inequalities were improved by finding out sharp missing terms and the best constants.(2) By making effective use of the improved Hardy-Sobolev inequalities, nonlinear problems were successfully studied.(3) By a linearization method the symmetry breaking of the Caffarelli-Khon-Nirenberg type inequalities were investigated to make clear its mechanism.(4) The operators on the Orlicz with a variable exponent p(x) were studied using the Hardy inequalities.

Report

(6 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report   Self-evaluation Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (31 results)

All 2012 2011 2010 2009 2008 Other

All Journal Article (19 results) (of which Peer Reviewed: 19 results) Presentation (9 results) Remarks (3 results)

  • [Journal Article] The Caffarelli-Kohn-Nirenberg inequalities involving critical and supercritical weights2012

    • Author(s)
      T. Horiuchi, P. Kumulin
    • Journal Title

      Proceedings of the Japan Academy

      Volume: Vol.88, Ser. A No.1 Pages: 1-6

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] The Caffarelli-Kohn-Nirenberg inequalities involving critical and supercritical weights2012

    • Author(s)
      T.Horiuchi, P.Kumlin
    • Journal Title

      Proceedings of the Japan Academy

      Volume: 88 Pages: 1-6

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Riesz transforms on generalized Hardy spaces and a uniqueness theorem for the Navier-Stokes equations2011

    • Author(s)
      E. Nakai, T. Yoneda
    • Journal Title

      Hkkaido Math. J.

      Volume: 40-1 Pages: 67-88

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Caloric morphisms between different radial metrics on semi-euclidean spaces of same dimension2011

    • Author(s)
      K.Shimomura
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 43 Pages: 13-41

    • NAID

      130001438928

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Liouville type theorem associated with the wave equation2011

    • Author(s)
      K.Shimomura
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 43 Pages: 51-64

    • NAID

      130001438930

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Boundedness of fractional integral operators on Morrey spaces and Sobolev embeddings for generalized Riesz potentials2010

    • Author(s)
      Y. Mizuta, E. Nakai, T. Ono, T. Simomura
    • Journal Title

      J. Math. Soc. Japan

      Volume: 62-3 Pages: 707-744

    • NAID

      10027870763

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Pointwise multipliers of Orlicz spaces2010

    • Author(s)
      L. Maligranda, E. Nakai
    • Journal Title

      Arch. Math.(Basel)

      Volume: 95-3 Pages: 251-256

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Singular and fractional integral operators on Campanato spaces with variable growth conditions2010

    • Author(s)
      E. Nakai
    • Journal Title

      Rev. Mat. Complut

      Volume: 23-2 Pages: 355-381

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Hiroshima Caloric morphisms for rotation invariant metrics2010

    • Author(s)
      K. Shimomura
    • Journal Title

      Math. J.

      Volume: 40-3 Pages: 315-331

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Singular and fractional integral operators on Campanato spaces with variable growth conditions2010

    • Author(s)
      中井英一
    • Journal Title

      Rev.Mat.Complut. Vol.23, No.2

      Pages: 355-381

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Caloric morphisms for rotation invariantmetrics2010

    • Author(s)
      Nishio, Masaharu shimomura, Katsunori
    • Journal Title

      Hiroshima Math.J. Vol.40 No.3

      Pages: 315-331

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Singular and fractional integral operators on Campanato spaces with_variable growth conditions2010

    • Author(s)
      中井英一
    • Journal Title

      Rev.Mat.Complut.

      Volume: Vol.23, No.2 Pages: 355-381

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Caloric morphisms for rotation invariant metrics2010

    • Author(s)
      Nishio, Masaharu Shimomura, Katsunori
    • Journal Title

      Hiroshima Math.J.

      Volume: Vol.40, No.3 Pages: 315-331

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Missing terms in the weighted Hardy-Sobolev inequalities and its application2010

    • Author(s)
      安藤広, 堀内利郎
    • Journal Title

      Journal of Mathematics of Kyoto Unuv.(accepted) (未定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Construction of solutions, for the initial value problem of a functional-differential equation of advanced type2009

    • Author(s)
      中井英一
    • Journal Title

      Aequationes Mathematicae Vol.77, No.3

      Pages: 259-272

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Missingterms in the weighted Hardy-Sobolev inequalities and its application2009

    • Author(s)
      安藤広、堀内利郎
    • Journal Title

      Journal of Mathematics of Kyoto Unuv. (accepted) (未定)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Kato's inequalities for degenerate quasilinear elliptic operators2008

    • Author(s)
      T. Horiuchi
    • Journal Title

      Kungpook Mathematical J.

      Volume: 48-1 Pages: 15-24

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Orlicz-Morrey spaces and the Hardy-Littlewood maximal function2008

    • Author(s)
      中井英一
    • Journal Title

      Studia Mathematica 188 No3

      Pages: 193-221

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Missing terms in the weighted Hardy-Sobolev inequalities and its application

    • Author(s)
      安藤広、堀内利郎
    • Journal Title

      Journal of Mathematics of Kyoto Unuv. accepted

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Presentation] Construction of slowly increasing functions2012

    • Author(s)
      安藤広、堀内利郎、中井英一
    • Organizer
      日本数学会
    • Place of Presentation
      東京
    • Year and Date
      2012-03-29
    • Related Report
      2011 Final Research Report
  • [Presentation] Construction of slowly increasing functions2012

    • Author(s)
      安藤広
    • Organizer
      日本数学会
    • Place of Presentation
      東京理科大
    • Year and Date
      2012-03-29
    • Related Report
      2011 Annual Research Report
  • [Presentation] On the Symmetry breaking of the non-critical Caffarelli-Khon-Nirenberg type inequalities by a linearization method2012

    • Author(s)
      T, Horiuchi
    • Organizer
      日本数学会
    • Place of Presentation
      東京
    • Year and Date
      2012-03-27
    • Related Report
      2011 Final Research Report
  • [Presentation] 重み付きハーディの不等式の精密化に付いて2012

    • Author(s)
      安藤広、堀内利郎、中井英一
    • Organizer
      日本数学会
    • Place of Presentation
      東京
    • Year and Date
      2012-03-27
    • Related Report
      2011 Final Research Report
  • [Presentation] On the symmetry breaking of the non-critical Caffarelli-Khon-Nirenberg inequalities by a linearization method2012

    • Author(s)
      堀内利郎
    • Organizer
      日本数学会
    • Place of Presentation
      東京理科大
    • Year and Date
      2012-03-27
    • Related Report
      2011 Annual Research Report
  • [Presentation] 重み付きハーディの不等式の精密化に付いて2012

    • Author(s)
      安藤広
    • Organizer
      日本数学会
    • Place of Presentation
      東京理科大
    • Year and Date
      2012-03-27
    • Related Report
      2011 Annual Research Report
  • [Presentation] On the Caffarelli-Kohn-Nirenberg type inequalities involving Critical and Supercritical weights2009

    • Author(s)
      T, Horiuchi
    • Organizer
      姫路研究集会
    • Place of Presentation
      姫路
    • Year and Date
      2009-02-21
    • Related Report
      2011 Final Research Report
  • [Presentation] On the Caffarelli-Kohn-Nirenberg type inequalities involving Critical and Supercritical weights2008

    • Author(s)
      T, Horiuchi
    • Organizer
      日本数学会
    • Place of Presentation
      東京
    • Year and Date
      2008-09-24
    • Related Report
      2011 Final Research Report
  • [Presentation] On the Caffarelli-Kohn-Nirenberg inequalities involving Critical and Supercritical weights2008

    • Author(s)
      堀内利郎
    • Organizer
      日本数学会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2008-09-24
    • Related Report
      2008 Annual Research Report
  • [Remarks]

    • URL

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

    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • URL

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

    • Related Report
      2009 Annual Research Report
  • [Remarks]

    • URL

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

    • Related Report
      2008 Annual Research Report

URL: 

Published: 2008-04-01   Modified: 2016-04-21  

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