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The research on degenerate elliptic partial differential equations by the method of real analysis

Research Project

Project/Area Number 11640150
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 Univ., Fuc. of Science, Prof., 理学部, 教授 (80157057)

Co-Investigator(Kenkyū-buntansha) ANDO Hiroshi  IBARAKI Univ., Fuc. of Science, Assistant, 理学部, 助手 (60292471)
SHIMOMURA Katsunori  IBARAKI Univ., Fuc. of Science, A. Prof., 理学部, 助教授 (00201559)
ONISHI Kuzuei  IBARAKI Univ., Fuc. of Science, Prof., 理学部, 教授 (20078554)
HORIHATA Kazuhiro  Tohoku Univ., mathematical Inst. Assistant, 理学研究科, 助手 (10229239)
NAKAI Eiichi  Osaka Educational Univ., Fuc. of Education, A. Prof., 教育学部, 助教授 (60259900)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2001: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2000: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1999: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordsdegeneration alliptic / Singularity / Sobolev inequality / Hardy inequality / P-harmonic / potential theory / non-linear eliptic / ハーディー不等式 / p調和方程式 / 非線形楕円型 / 除去可能な特異点 / ソボレフの不等式 / ハーディーの不等式 / 非線型楕円型方程式 / 特異点の除去可能性 / うめ込み定理 / 解の正則性
Research Abstract

1. On the Removable singularities of solutions for degenerate ellptic equations:
(1) When the principal part of the operator is linear, we have succeeded in establishing a sufficient condition in order to remove singularities of solutions of degenerate elliptic equations with absorption terms. Moreover, if the set on which the operator may be degenerate is smooth enough, then this condition is also necessary.
(2) We have extended our results on the linear operators to the case when the principal part is nonlinear. In order to overcome the difficulties caused from nonlinearlity, we used the conjugate function (Legendre transform) of the absorption term and modified capacities.
2. On the regularity of solutions for genuinely degenerated elliptic equations: The existence of bounded solutions for degenerate elliptic equations were studied. To study further regularities, multiplicative Sobolev inequalities with weights were established.
3. On the quasilinear elliptic equations with critical nonlinear terms: Quasilinear elliptic equations with critical nonlinear terms were successfully investigated, and as an application, the best constants for various types of weighted Sobolev inequalities with weights were determined. We also showed that the best constants and the existence of extremal functions essentially depend on the degeneracy of the operators in a very subtle way.
4. Missing terms in the generalized Hardy inequalities were investigated in connection with 3.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (36 results)

All Other

All Publications (36 results)

  • [Publications] 堀内利郎: "Removable singubrities for quasilinean degenerate elliptic equations with absurption term"J. Math. Soc. Japan. 53-3. 513-540 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 堀内利郎: "Removable singularities for semilinear degenerate elliptic equations and its applications"J. Math.of Kyoto Univ.. 40-1. 13-35 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 堀内利郎: "Some remarks on Kato's inequality"J. of inequalities and applications. 6. 29-36 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 島倉紀夫: "Normal Coordinate systems from a viewpoint of real analysis"Tohoku Math. J (2). 52-4. 533-553 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 保城寿彦: "Mourre's method and smoothing properties of dispersive equations"Comm. Math. Phys.. 202. 255-265 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 下村勝孝: "The determination of caloric morphisms on Euclidean demains"Nagoya Math. J.. 158. 133-166 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Horiuchi, Toshio.: "Removable singularities for quasilinear degenerate elliptic equations with absorption term"J. Math. Soc. Japan. Vol. 53, No. 3. 513-540 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Horiuchi, Toshio.: "Removablesingularities for semilinear degenerate elliptic equations and its applications"J. Math, of Kyoto Univ.. Vol. 40, No. 1. 13-35 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Horiuchi, Toshio.: "Some remarks on Kato's inequality"Journal of inequalities and applications. Vol. 6. 29-36 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Horiuchi, Toshio.: "Quasilinear degenerate elliptic equations with absorption term"Proc. Japan Acad.. 75, Ser. A.. 122-1251 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shimakura, Norio.: "Normal coordinate systems from aviewpoint of real analysis."Tohoku Math. J. (2). 52, no. 4. 533-553 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hoshiro, Toshihiko.: "Mourre's method and smoothing properties of dispersive equations."Comm. Math. Phys.. 202, no. 2. 255-265 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hoshiro, Toshihiko.: "On the estimates for Helmholtz operator."Tsukuba J. Math.. 23, no. 1. 131-149 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shimomura, Katsunori.: "The determination of caloric morphisms on Euclidean domains."Nagoya Math. J.. 158. 133-166 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Nishio, Masaharu; Shimomura, Katsunori; Suzuki, Noriaki: "Note on poly-super temperatures on a strip domain."Osaka J. Math. 36, no. 3. 539-556 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Horihata, Kazuhiro.: "The evolution of harmonic maps."Tohoku Mathematical Publications 11. Tohoku University, Mathematical Institute, Sendai,. iv+111pp. Dissertation. (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] E. Nakai and Hironori Sumitomo: "On Generalized Riesz potentials and spaces of some smooth functions"Scientiae Mathematicae. Vol. 4. 891-900 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eichi. Nakai.: "On Generalized fractional integrals in the Orlicz spaces on spaces of Homogeneous type"Scientiae Mathematicae. Vol. 4. 901-915 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 堀内 利郎: "Removable singularities for semilinear degenerate elliptic equations and its applications"J. Math. Of Kyoto Univ.. 40・1. 13-35 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] 堀内 利郎: "Removable singularities for quasilinear degenerate elliptic equations with absorption term"J. Math. Soc. Japan. 53・3. 513-540 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 堀内 利郎: "Some remarks on Kato is inequality"Journal of inequalities and applications. 6. 29-36 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] 中井 英一: "A characterization of pointwise multipliers on the Morrey spaces"Scientiae Mathematicae. 3・3. 445-454 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] 中井 英一: "On generalized fractional integrals"Taiwanese J. of Mathematics. (未定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 島倉 紀夫: "Huygens operators on product manifolds"Tohoku Math. J.. (未定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 堀内利郎: "Removable singularities for semilinear degenerate elliptic equations and its applications"J.Math.of Kyoto Univi.. 40・1. 13-35 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 堀内利郎: "Some remarks on Kato's inequatity"J.of Inequalities and applications. (未定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 堀内利郎: "Removable singularities for quasilinear degenerate elliptic equations with absorption term"J.of Math.Soc.of Japan. (未定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 堀内利郎: "The potential theory for degenerate elliptic operators"Kyungpook Mathematical Journal. 40・1. 125-138 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 堀内利郎: "The Neumann boundary value problem for degenerate elliptic operators II"Far east Journal of Math.Sciences. 2・2. 235-250 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 堀内利郎: "The Neumann boundary value problem for degenerate elliptic operators III"Far east Journal of Math.Sciences.. 2・3. 345-363 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 堀内 利郎: "Some remarks on Kato's inequality"J.of Inequalities and applications. (未定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 堀内 利郎: "Quasilinear degenerate elliptic equations with absorption term"Proc.Japan Acad.75 ser.A.. 75 A. 122-125 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 堀内 利郎: "The potential theory for degenerate elliptic operators"Kyungpook Mathematical Journal. (未定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 堀内 利郎: "The Neumann boundary value problem for degenerate elliptic operators"Far East J.of Mathematical Sciences. 2-1. 7-26 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 堀内 利郎: "The Neumann boundary value problem for degenerate elliptic operators II"Far East J.of Mathematical Sciences. 2-2. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 堀内 利郎: "Removable singlurities for semilinear degenerate elliptic equations and its applications"J.Math.of Kyoto Univ. 40-1(未定). (2000)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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