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Qualitative studies of solutions to elliptic equations in unbounded domains

Research Project

Project/Area Number 09640192
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionHiroshima University

Principal Investigator

USAMI Hiroyuki  Hiroshima University Faculty of Integrated Arts & Sciences, assistant professor, 総合科学部, 助教授 (90192509)

Co-Investigator(Kenkyū-buntansha) NAITO Yuki  Kobe University Faculty of Engineering assistant professor, 工学部, 助教授 (10231458)
NAKAYAMA Hiromichi  Hiroshima University Faculty of Integrated Arts & Sciences, assistant professor, 総合科学部, 助教授 (30227970)
SHIBATA Tetsutaro  Hiroshima University Faculty of Integrated Arts & Sciences, assistant protessor, 総合科学部, 助教授 (90216010)
NAITO Manabu  Ehime University Faculty of Science, Professor, 理学部, 教授 (00106791)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 1998: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1997: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordselliptic equation / degenerate Laplacian / Liouville-type theorem / Sturm-Liouville problem / oscillation / eigenvalue problem / symmetry of solutions / elliptic system / 非線形楕円型方程式 / リューヴィユ型定理 / 非線形固有値問題
Research Abstract

(1) Oscillation criteria for elliptic equations : Effective oscillation criteria are established for second-order quasilinear elliptic equations whose leading terms are degenerate Laplacians. Our method is based on comparison principles and asymptotic theory for quasilinear ordinary differential equations. For one-dimensional case, useful information about numbers of zeros of solutions is obtained via the generalized Prufer transformation.
(2) Liouville-type theorems and nonexistence of positive solutions of BVPs : Lioville-type theorems are established for quasilinear elliptic equations whose leading terms are degenerate Laplacians and (generalized) mean curvature operators. Our results can be regarded as a natural extension of classical Liouville theorem.
(3) Symmetry of positive solutions for elliptic problems : We prove by means of the moving plane method or moving sphere method that positive solutions of elliptic equations of certain types are radially symmetric. Useful information about self-similar solutions of parabolic problems can be derived from our results.
(4) Two-parameter eigenvalue problems : Two-parameter nonlinear Sturm-Liouville problems are considered. The existence of the variational eigenvalues is established. Asymptotic formulas of eigenvalues and eigenfunctions are obtained.
(5) Asymptotic theory for solutions of elliptic systems : Semilinear elliptic systems are considered. We establish nonexistence criteria of positive solutions, Liouville-type theorems, and oscillation criteria.)

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] H.Usami: "Some oscillation theorems for a class of quasilinear elliptic equations" Ann.Mat.Pura.Appl.175. 277-283 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Elbert: "Singular eigenvalue problems for second order linear ordinary differential equations" Arch.Math.(Brno). 34・1. 59-72 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] S.Matsumoto: "On ν-distal flows on 3-manifolds" Bull.London Math.Soc.29. 609-616 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Shibata: "Asymptotic profiles of variational eigenvalues of two-parameter non-linear Sturm-Liouville problems" Math.Meth.Appl.Sci.21. 1619-1635 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Naito: "Radial symmetry of positive solutions for semilinear elliptic equations on the unit ball in R^n" Funkcial.Ekvac.41・2. 215-234 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusahara: "A barrier method for quasilinear ordinary differential equations of the curvature type" Czechoslovak Math.J.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Usami: "Some oscillation theorems for a class of quasillnear elliptic equations" Ann.Mat.Pura Appl.175. 277-283 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Elbert et al: "Singular eigenvalue problems for second order linear ordinary differential equations" Arch.Math. (Brno). 34-1. 59-72 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] S.Matsumoto et al: "On nu-distal flows on 3-manifolds" Bull.London math.Soc.29. 609-616 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Shibata: "Asymptotic profiles of variational eigenvaluse of two-parameter non-linear Sturm-Liouville problems" Math.Meth Appl.Sci.21. 1619-1635 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Maito et al: "Redial symmetry of positive solutions for semilinear elliptic equations on the unit ball in R^n" Funkcial.Ekvac.41-2. 215-234 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusahara et al: "A barrier method for quasilinear ordinary differential equations of the curvature type" Czechoslovak Math.J.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Usami: "Some oscillation theorems for a class of quasilinear elliptic equations" Ann.Mat.Pura.Appl.175. 277-283 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Elbert: "Singular eigenvalue problems for second order linear ordinary differential equations" Arch.Math.(Brno). 34・1. 59-72 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Naito: "A note on the moving sphere method" Pacific J.Math.

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Shibata: "Asymptotic profiles of variational eigenvalues of two-parameter non-linear Sturm-Liouville problems" Maath.Meth.Appl.Sci.21. 1619-1635 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Naito: "Radial symmetry of positive solutions for semilinear elliptic equations on the unit ball in R^n" Funkcial.Ekvac.41・2. 215-234 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Kusahara: "A barrier method for quasilinear ordinary differential equations of the curvature type" Czechoslovak Math.J.

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Usami: "Some oscillation theorems for a class of quasilinear elliptic equations" Ann.Mat.Pura Appl.(発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Shibata: "Nonlinear multiparameter eigenvalue problems on general level sets" Nonlinear Anal.29. 823-838 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Shibata: "Spectral asymptotics for nonlinear non-autcnomous elliptic eigenvalue problems in a ball" Applicable Analysis. 64. 379-390 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] S.Matsumoto: "On V-distal flows on 3-manifolds" Bull.London Math.29. 609-616 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Kusano: "A singular eigenvalue problem for Sturm-Liouville eguations" Differentsial'nye Uravneniya. (発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Naito: "Entire solutions of the in eqality div(A(lDul)Du)≧f(u)" Math.Z. 225. 167-175 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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