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the positivity of degenerate elliptic operators and the microlocal analysis on solutions for partial differentiai equations

Research Project

Project/Area Number 12440038
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYOTO UN IVERSITY

Principal Investigator

MORIMOTO Yoshinori  Kyoto Univ., Graduate School of Human and Environmental Studies, Professor, 大学院・人間・環境学研究科, 教授 (30115646)

Co-Investigator(Kenkyū-buntansha) TARAMA Shigeo  Osaka City Univ., Faculty of Technology, Professor, 工学部, 教授 (90115882)
UEKI Naomasa  Kyoto Univ., Graduate School of Human and Environmental Studies, Ass. Professor, 大学院・人間・環境学研究科, 助教授 (80211069)
TAKASAKI Kanehisa  Kyoto Univ., Graduate School of Human and Environmental' Studies, Professor, 大学院・人間・環境学研究科, 教授 (40171433)
ANDO Hiroshi  Ibaraki Univ., Faculty of Science, Assistant, 理学部, 助手 (60292471)
MORIOKA Tatsushi  Osaka Educational Univ., Faculty of Education, Ass. Professor, 教育学部, 助教授 (80239631)
畑 政義  京都大学, 総合人間学部, 助教授 (40156336)
浅倉 史興  大阪電気通信大学, 工学部, 教授 (20140238)
Project Period (FY) 2000 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥8,700,000 (Direct Cost: ¥8,700,000)
Fiscal Year 2003: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2002: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2001: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2000: ¥2,500,000 (Direct Cost: ¥2,500,000)
Keywordsinfinitely degenerate / microlocal analysis / Sobolev inequality of logarithmic type / semiliner Dirichlet problem / Fefferman-Phong inegality / Schrodinger equation / hypaellipticity / Wick calculus / 退化Monge-Ampere方程式 / 対数型ソボレフ不等式 / 最大値原理 / Schodinger方程式 / key wards / Wick作用素 / 楕円型方程式 / 双極性 / 対数型 / sololev不等式 / 半線型楕円型方程式 / schodinger方程式 / 強一意性定理 / 双曲型方程式
Research Abstract

The purpose of this research is to study how the positivity of degenerate elliptic operators is reflected to the structure of solutions for partial differential equations, by using the theories of pseudo-differential operators, Fourier integral operators, harmonic analysis and stochastic calculus. Head investigator considered the Dirichlet problem for certain semilinear elliptic equations whose principal parts of second order degenerate infinitely, by joint research with Prof. Xu who is a foreigner joint research person. Firstly, the existence and the boundedness of solutions to this problem were shown, and secondly the continuity and C∞ regularity of solutions were clarified. The logarithmic regularity up estimate can be only expected for certain infinitely degenerate elliptic operators with weak positivity, differing from the case for elliptic operators with finite degeneracy. Under the assumption of this logarithmic regularity up estimate, we derived the Sobolev inequality of logari … More thmic type, and proved the existence of solutions to the Dirichlet problem by solving the associated variational problem. The proofs of the boundedness, the continuity and C∞ regularity of solutions to our problem are completely different from the traditional methods used for semilinear equations whose principal part is elliptic or sub-elliptic. Our method is based on the technique for C∞-hypoellipticity for linear infinitely degenerate elliptic operators. In relation to the positivity of degenerate elliptic operators, the recent results of J.-M.Bony and D.Tataru were examined, where the inequality of Fefferman-Phong concerning the positivity of pseudodifferential operators are discussed. As a joint research with Prof. Lerner who introduced firstly Wick calculus for the research of solvability of pseudodifferential operators of principal type, we showed that the Wick calculus is also applicable to the proof of Fefferman-Phong inequality instead of FBI operators employed in Tataru's paper. Our another proof is carried out in refining the product formula of Wick operators obtained in the joint work with Ando. An investigator Ueki studied the spectrum of a Schrodinger operator with the random magnetic field relevant to the microlocal analysis with infinitely degeneracy, found out that a density-of-states function have remarkably different structure in the case of Pauli Hamiltonian from the former case, and applied those results to research of the hypoellipticity for ∂b-Laplacian. From the point of view on the microlocal analysis for partial differential equations, the Goursat problem to the second order equation was considered by an investigator Tarama who extended Hasegawa's result by energy estimates, and the algebraic geometry structure of the particular solution to soliton equations was studied by an investigator Takasaki, in relation to the singular solutions for degenerate elliptic equations. Less

Report

(5 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (33 results)

All Other

All Publications (33 results)

  • [Publications] 森本芳則, C-J.Xu: "Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators"Asterisque. 284. 245-264 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 森本芳則, C-J.Xu: "Regularity of weak solution for a class of infinitely degenerate ellitpic semilinear equations"Seminaire Equations aux Derivees Partielles Ecole Polytechnique. 03-04. VII-1-VII-14 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Lerner, 森本芳則: "An aspect on the positivity of pseudodifferential operators via Wick calculus"(未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 上木直昌: "Estimates on the heat kernel of the Pauli Hamiltonian and its application to problems on hypoellipticity of the <∂_b>^^^-Laplacian"Math.Z.. 239. 69-97 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 多羅間茂雄: "On the Goursat problem for a second order equation"Electron.J.Diff.Eqns.. 2002・52. 1-28 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 高崎金久: "An integral system on the moduli space of rational functions and its variants"Journal of Geometry and physics. 47. 1-20 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yoshinori Morimoto, Chao-Jiang Xu: "Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators"Asterisque. Vol.284 (in honor of J.-M.Bony). 245-264 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yoshinori Morimoto, Chao-Jiang Xu: "Regularity of weak solution for a class of infinitely degenerate ellitpic semilinear equations,"Seminaire Equations aux Derivees Partielles Ecole Polytechnique. VII-1-VII-14 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Nicolas Lerner, Yoshinori Morimoto: "An aspect on the positivity of pseudodifferential operators via Wick calculus"(preprint).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Naomasa Ueki: "Estimates on the heat kernel of the Pauli Hamiltonian and its application to problems on hypoellipticity of the ∂b -Laplacian"Math. Z. 239(1). 69-97 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Shigeo Tarama: "On the Goursat problem for a second order equation"Electron. J. Diff. Eqns.. Vol. 2002, No.52. 1-28 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kanehisa Takasaki, Takebe: "An integral system on the moduli space of rational functions and its variants"Journal of Geometry and Physics. 47. 1-20 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 森本芳則, C-J.Xu: "Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators"Asterisque. 284. 245-264 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 森本芳則, C-J.Xu: "Nonlinear hypoellipticity for a class of infinitely degenerate elliptic operators"Colloque Tuniso-Francais a Hamammet, 16-21 Septembre 2003. (発表予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] 森本芳則, C-J.Xu: "Regularity of weak solution for a class of infinitely degenerate ellitpic semilinear equations"Seminaire Equations aux Derivees Partielles Ecole Polytechnique.2003-2004. VII-1-VII-14 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Lerner, 森本芳則: "An aspect on the positivity of pseudodifferential operators via Wick calculus"数理研講究録. (未定).

    • Related Report
      2003 Annual Research Report
  • [Publications] 高崎金久: "An integral system on the moduli space of rational functions and its variants"Journal of Geometry and Physics. 47. 1-20 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 森本芳則: "Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators"Asterisqueo Special Vol. in honor of Jean Michel Bony. (未定). 245-264 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 森本芳則: "Wick calculus and the Cauchy problem for some dispersive equations"Osaka J. Math.. 31巻. 123-147 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Shigeo Tarama: "On the Goursat problem for a second order equation"Electron J. Diff. Egns.. 2002巻52号. 1-28 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 高崎金久: "Integrable systems whose spectral curve is the graph of a function"未定. (未定).

    • Related Report
      2002 Annual Research Report
  • [Publications] 上木直昌: "Lifschitz tail of a random Pauli Hamiltonian with an anomalous magnetic moment"Japanese Journal of Mathematics. 28巻・2号. 261-286 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 畑 政義: "Pade approximation to the logarithmic derivative of the Gauss hypergeometric, function"Analytic Number Theory, Developments in Mathematics. 6巻. 157-172 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 森本芳則: "Wick calculus and the Cauchy problem for some dispersive equations"Osaka J.Math.. 39巻・1号(未定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 上木直昌: "Estimates on the heat Kernel of the Paul Hamiltonian and its application to problems on hypoellipticity of the 2b-Laplacian"Math.Z.. 239巻1号. 69-97 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 畑政義: "Pade approximation to the logarithmic derivative of the Gauss hypergeometric function"Proc. Conf. on Analytic Number Theory. 6巻(未定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 森本芳則: "Remark on the analytic smoothing for the Schrodinger equation"Indiana Univ.Math.. (未定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 畑政義: "C^2-saddle method and Benkers' integral"Trans,Amer,Math.Soc.. 352巻. 4557-4583 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 上木直昌: "Simple examples of Lifschitz tails in Gaussian random magnetic fields"Ann.H.Poincare. 1巻3号. 473-498 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 上木直昌: "Estimates on the heat kernel of the Pauli Hamiltonian and its application to problems on hypoellipticity of the ∂_b-Laplacian"Math.Z.. (未定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 多羅間茂雄: "The Cauchy and Mixed problems for semilinear wave equations with nonlinear damping terms"Math.Japonica. 51巻3号. 459-466 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 多羅間茂雄: "On the estimate of some conjugation"Mem.Fac.Eng.Osaka City Univ.. 41巻. 117-123 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 森岡達史: "第2超局所解析の基本"龍谷大学科学技術共同研究センター(未定).

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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