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Refinement of classical inequalities and its application to elliptic variational problems

Research Project

Project/Area Number 24540157
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
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) NAKAI EIICHI  茨城大学, 理学部, 教授 (60259900)
SHIMOMURA KATUNORI  茨城大学, 理学部, 教授 (00201559)
ANDO HIROSHI  茨城大学, 理学部, 講師 (60292471)
大西 和榮  茨城大学, 理学部, 教授 (20078554)
Co-Investigator(Renkei-kenkyūsha) HOSHIRO TOSHIHIKO  兵庫県立大学, 物質理学研究科, 教授 (40211544)
SATO TOKUSHI  東北大学, 理学研究科, 助教 (00261545)
Project Period (FY) 2012-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
KeywordsCKN型不等式 / ハーディ不等式 / ソボレフ不等式 / 楕円型変分問題 / ミッシング・ターム / p-ラプラシアン / 加藤の不等式 / 強最大値原理 / CKN型不等式 / ソボレフの不等式 / ハーディの不等式 / 対称性の崩れ / 非線形楕円型方程式 / Super logarithm / CKN 型不等式 / 重み付きソボレフの不等式 / 重み付きハーディの不等式 / p-ラプラシアン / ミッシンク・ターム / Hardy-Sobolevの 不等式 / Hardyの不等式の精密化 / スーパー対数関数 / 不等式の精密化 / 超対数関数 / 不等式のミッシングターム / 退化型非線形偏微分方程式
Outline of Final Research Achievements

(1) Existence of the solutions for the best constants of the Caffarelli-Kohn-Nirenberg type inequalities, continuity with respect to parameters of the best constant, symmetry breaking were studied systematically. For p=1, the Caffarelli-Kohn-Nirenberg type inequalities were established by effective use of isoperimetric inequalities, and symmetry breaking in p=1 was proved. (2) A theory of super logarithm based on logarithmic infinite product was introduced, and existence of infinitely many missing term was shown for the weighted Hardy type inequalities. (3) Differential equations characterizing super logarithm was studied to deal with elliptic equations with nonlinear term of very fast growth order.

Report

(5 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (14 results)

All 2016 2015 2014 2012 Other

All Int'l Joint Research (1 results) Journal Article (10 results) (of which Peer Reviewed: 10 results,  Open Access: 2 results,  Acknowledgement Compliant: 2 results) Presentation (3 results)

  • [Int'l Joint Research] シャルマース工科大学(スウェーデン)

    • Related Report
      2015 Annual Research Report
  • [Journal Article] radial symmetry and its breaking in the Caffarelli-Kohn-Nirenberg type inequalities for p=12016

    • Author(s)
      N. Chiba, T. Horiuchi
    • Journal Title

      Proceedings of the Japan academy Series A Mathematical Sciences

      Volume: 92 Pages: 51-55

    • NAID

      130005100042

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Caffarelli-Kohn-Nirenberg 型不等式とその周辺2016

    • Author(s)
      堀内利郎
    • Journal Title

      数学(日本数学会)

      Volume: 68 Pages: 1-23

    • NAID

      130006321022

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On radial symmetry and its breaking in the Caffarelli-Kohn-Nirenberg type inequalities for p=12015

    • Author(s)
      N. Chiba, T. Horiuchi
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 47 Pages: 49-63

    • NAID

      130005100042

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Weighted Hardy inequalities with infinitely many sharp missing terms2014

    • Author(s)
      Ando Hiroshi, Horiuchi Toshio, Nakai eiichi
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 46 Pages: 9-30

    • NAID

      130004697044

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Some properties of slowly increasing functions2014

    • Author(s)
      Ando Hiroshi, Horiuchi Toshio, Nakai eiichi
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 46 Pages: 37-49

    • NAID

      130004697043

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Weighted Hardy inequalities with infinitely many sharp missing terms2014

    • Author(s)
      Ando, Hiroshi; Horiuchi, Toshio; Nakai
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 48 Pages: 9-29

    • NAID

      130004697044

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Caffarelli-Kohn-Nirenberg inequalities involving critical and supercritical weights2012

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

      Kyoto Journal of Mathematics

      Volume: 52(4) Pages: 661-742

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Missing terms in the weighted Hardy-Sobolev inequalities and its application2012

    • Author(s)
      H. Ando, T. Horiuchi
    • Journal Title

      Kyoto Journal of Mathematics

      Volume: 52(4) Pages: 759-796

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] CONSTRUCTION OF SLOWLY INCREASING FUNCTION2012

    • Author(s)
      H. Ando, T. Horiuchi, E. Nakai
    • Journal Title

      Scientiae Mathematicae Japonicae Online

      Volume: 25 Pages: 207-221

    • NAID

      130007663411

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the weighted rearrangement of functions and Degenerate Nonlinear elliptic equations2012

    • Author(s)
      H. Ando, T. Horiuchi
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 44 Pages: 17-31

    • NAID

      130003373916

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] Radial symmetry and its breaking in the Caffarelli-Kohn-Nirenberg type inequalities for p = 12015

    • Author(s)
      堀内利郎、千葉奈緒希
    • Organizer
      日本数学会(函数方程式分科会)
    • Place of Presentation
      京都産業大学
    • Year and Date
      2015-09-13
    • Related Report
      2015 Annual Research Report
  • [Presentation] p-ラプラシアンを含む強最大値原理についての注意2015

    • Author(s)
      劉暁静、堀内利郎
    • Organizer
      日本数学会(函数方程式分科会)
    • Place of Presentation
      京都産業大学
    • Year and Date
      2015-09-13
    • Related Report
      2015 Annual Research Report
  • [Presentation] Recent progress in Caffarelli-Kohn-Nirenberg inequalities2014

    • Author(s)
      Horiuchi Toshio
    • Organizer
      ICM 2014( International congress of Mathematician 2014 )
    • Place of Presentation
      Seoul Korea
    • Year and Date
      2014-08-21
    • Related Report
      2014 Research-status Report

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Published: 2013-05-31   Modified: 2019-07-29  

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