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Potential theoretic approach to parabolic equations

Research Project

Project/Area Number 13640186
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionOsaka City University

Principal Investigator

NISHIO Masaharu  Osaka City University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90228156)

Co-Investigator(Kenkyū-buntansha) KOMATU Takashi  Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80047365)
SAKAN Ken-ichi  Osaka City University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70110856)
MASAOKA Hiroaki  Kyoto Sangyo University, School of Science, Professor, 大学院・理学研究科, 教授 (30219315)
SUZUKI Noriaki  Nagoya University, Graduate School of Mathematics, Associate Professor, 理学部, 教授 (50154563)
SHIMOMURA Katsunori  Ibaraki University, School of Science, Associate Professor, 理学部, 助教授 (00201559)
今吉 洋一  大阪市立大学, 大学院・理学研究科, 教授 (30091656)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Keywordsheat equation / mean value property / harmonic function / quasiconformal mapping / Martin boundary / caloric morphism / polytemperature / parabolic equation of fractional order / 放物型方程式 / ポテンシャル論 / ベルグマン空間 / 再生核 / 掃散測度 / ヘルグマン空間 / カルレソン測度 / ペアノ核 / グリーン関数 / 最大値の原理 / クリーン関数
Research Abstract

We have studied on parabolic equations and related subjects to obtain the following results :
1. Mean value property.
Suzuki characterized the heat ball by using mean value properties. He had a result on density functions of mean value properties.
2. Quasiconformal mappings.
(1) Sakan improved the Heinz's results for harmonic and quasiconformal mappings
(2) Masaoka showed that the rigidity property of the minimal Martin boundary of Rieman surfaces of Heinz type for quasiconformal mappings.
3. Martin boundary.
Masaoka gave the relation of various classes of harmonic functions with Martin boundary of Rieman surfaces of Heinz type.
4. Caloric morphism
(1) Nishio and Shimomura gave a differential equation which characterizes caloric morphisms between manifolds.
(2) Shimomura classifed caloric morphisms on a certain semi-riemannian manifold.
5. Polytemperature.
Shimomura showed that the Appell transformation is the only map which presearves polytemperatures.
6. Parabolic equation of fractional order
Nishio, Shimomura and Suzuki introduced the notion of parabolic Bergman spaces and obtained the fundamental results on completeness, reproducing kernels, dual spaces, etc.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (45 results)

All 2005 2004 2003 2002 2001 Other

All Journal Article (29 results) Publications (16 results)

  • [Journal Article] α-palabolic Bergman spaces2005

    • Author(s)
      M.Nishio, K.Shimomura, N.Suzuki
    • Journal Title

      Osaka J. Math. 42(印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] α-parabolic bergman spaces2005

    • Author(s)
      M.Nishio, K.Shimomura, N.Suzuki
    • Journal Title

      Osaka J.Math., vol.42 (to appear)

    • NAID

      120005986891

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Quasiconformal mappings and minimal Martin boundary of $p$-sheeted unlimited covering surfaces of the complex plane2005

    • Author(s)
      H.Masaoka, S.Segawa
    • Journal Title

      Kodai J.Math., (to appear)

    • NAID

      130003574489

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On an asymptotically sharp variant of Heinz's inequality2005

    • Author(s)
      D.Partyka, K.Sakan
    • Journal Title

      Ann.Acad.Sci.Fenn., (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] α-parabolic Bergman spaces2005

    • Author(s)
      M.Nishio, K.Shimomura, N.Suzuki
    • Journal Title

      Osaka Journal of Mathematics 42(印刷中)

    • NAID

      120005986891

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Kuramochi boundary and harmonic functions with finite Dirichlet integral2004

    • Author(s)
      N.Jin, H.Masaoka
    • Journal Title

      Osaka Journal of Mathematics 41

      Pages: 1-17

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A Characterization of calboric morphisims between manifolds2003

    • Author(s)
      M.Nishio, K.Shimomura
    • Journal Title

      Ann. Acad. Sci. Fenn. Math. 28

      Pages: 111-122

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Mean value densities for temparatures2003

    • Author(s)
      N.Suzuki, N.A.Watson
    • Journal Title

      Colloq. Math. 98

      Pages: 87-96

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A characterization of caloric morphisms between manifolds2003

    • Author(s)
      M.Nishio, K.Shimomura
    • Journal Title

      Ann.Acad.Sci.Fenn.Math., vol.28

      Pages: 111-122

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Mean value densities for temperatures2003

    • Author(s)
      N.Suzuki, N.A.Watson
    • Journal Title

      Colloq.Math., vol.98

      Pages: 87-96

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Harmonic functions on finitely sheeted unlimited covering surfaces2003

    • Author(s)
      H.Masaoka, S.Segawa
    • Journal Title

      J.Math.Soc.Japan, vol.15

      Pages: 323-334

    • NAID

      10011796503

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the Malliavin calculus for SDE's on Hilbert spaces2003

    • Author(s)
      T.Komatsu
    • Journal Title

      Acta Applicandae Mathematicae, vol.78

      Pages: 223-232

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Caloric morphisms with respect to radial metrics on Rn-{0}2003

    • Author(s)
      K.Shimomura
    • Journal Title

      Math.J.Ibaraki Univ., vol.35

      Pages: 35-53

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Calboric morphisims on semi-euclidean space2002

    • Author(s)
      M.Nishio, K.Shimomura
    • Journal Title

      Rev. Roumaine Math. Pures Appl. 47

      Pages: 727-736

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Q-compactification of harmonic spaces and the Choquet simplex2002

    • Author(s)
      T.Ikegami, M.Nishio
    • Journal Title

      Osaka J.Math. vol.39

      Pages: 931-944

    • NAID

      120004845782

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Caloric morphisms on semi-euclidean space2002

    • Author(s)
      M.Nishio, K.Shimomura
    • Journal Title

      Rev.Roumaine Math.Pures Apll., vol.47

      Pages: 727-736

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On Heinz's inequality2002

    • Author(s)
      D.Partyka, K.Sakan
    • Journal Title

      Bull.Soc.Sci.Lett.Lodz vol.52

      Pages: 27-34

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the Malliavin calculus for Markov processes with jumps2002

    • Author(s)
      T.Komatsu
    • Journal Title

      Probability Theory, Mathematical Statistics, (Proceedings of Ukrainian Mathematical Congress 2001), Kyiv,

      Pages: 59-72

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A characterization of heat balls by a mean value property for temperatures2001

    • Author(s)
      N.Suzuki, N.A.Watson
    • Journal Title

      Proc.Amer.Math.Soc., vol.129

      Pages: 2709-2713

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On pseudo-metrics on the space of generalized quasisymmetric Automorphisms of a Jordan curve2001

    • Author(s)
      D.Partyka, K.Sakan
    • Journal Title

      Ann.Univ.Mariae Curie-Sklodowska Sect.A, vol.55

      Pages: 115-138

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Simplified probabilistic approach to Hormander theorem2001

    • Author(s)
      T.Komatsu, A.Takeuchi
    • Journal Title

      Osaka J.Math., vol.38

      Pages: 681-691

    • NAID

      120004844750

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the smoothness of pdf of solutions to SDE of jump type2001

    • Author(s)
      T.Komatsu, A.Takeuchi
    • Journal Title

      International J.Differential Equations and Appl., vol.2

      Pages: 141-197

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Quasicnformal mappings and minimal Martin boundary of p-sheeted unlimited corering surfaces of the complex plane

    • Author(s)
      H.Masaoka, S.Segawa
    • Journal Title

      Kodai Math. J. (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On An asymtotically sharp variant of Heinz's inequality

    • Author(s)
      D.Portyka, K.Sakan
    • Journal Title

      Ann. Acad. Sci. Fenn. Math. (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Generalized Hormander theorem for non-local operators

    • Author(s)
      T.Komatsu, A.Takeuchi
    • Journal Title

      Proceedings of the First Sino-German Conference on Stochastic Analysis, World Scientific, (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the partial hypoellipticity of SDE's on Hilbert spaces

    • Author(s)
      T.Komatsu
    • Journal Title

      Proceedings of the International Summer Seminar "Stocastic Dynamical System", (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On transformations which preserve poly-temperatures of degree m

    • Author(s)
      K.Shimomura
    • Journal Title

      Math.J.Ibaraki Univ., vol.33

    • NAID

      110000076176

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Quasiconformal mappings and minimal Martin boundary of p-sheeted unlimited covering surfaces of the complex plane

    • Author(s)
      H.Masaoka, S.Segawa
    • Journal Title

      Kodai Mathematical Journal (発表予定)

    • NAID

      130003574489

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On an asymptotically sharp variant of Heinz's inequality

    • Author(s)
      D.Partyka, K.Sakan
    • Journal Title

      Ann.Acad.Sci.Fenn. (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] M.Nishio, K.Shimomura: "A characterization of caloric morphisms between manifolds"Ann.Acad.Sci.Fenn.Math.. 28. 111-122 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Suzuki, N.A.Watson: "Mean value densities for temperatures"Colloquium mathematicum. 98・1. 87-96 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Jin, H.Masaoka: "Kuramochi boundary and harmonic functions with finite Dirichlet integrals on unlimited covering surfaces"Osaka J.Math.. 141(発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Masaoka, S.Segawa: "Harmonic functions on finitely sheeted unlimited covering surfaces"J.Math.Soc.Japan. 55. 323-334 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Shimomura: "Caloric morphisms with respect to radial metrics on R^n\{0}"Math.J.Ibaraki Univ.. 35. 35-53 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Ikegami, M.Nishino: "Q-compactification of harmonic spaces and the Choquet simplex"Osaka Journal of Mathematics. 39・4. 931-944 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Nishino, K.Shimomura: "A characterization of caloric morphisms between manifolds"Ann. Acad. Sci. Fenn. Math.. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] D.Partyka, K.Sakan: "On Heinz's inequality"Bull. Soc. Sci. Letters. Lodz. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] G.Nakamura, N.Suzuki: "Polynomial solutions to boundary-value problems of the heat equation"Bull. Aichi. Inst. Tech.. 37. 33-38 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Suzuki: "Mean value property for temperatures on an annulus domain"RIMS kokyuroku. 1293. 168-174 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Masaoka, S.Segawa: "Harmonic function on finitely sheeted unlimited covering surfaces"Journal of the Mathematical Society of Japan. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Ikegami, M.Nishio: "Q-compactification of harmonic spaces and the Choquet simplex"Osaka Journal of Mathematics. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] D.Partyka, K.Sakan: "On pseudo-metrics on the space of generalized quasisymmetric automorphisms of a Jordan curve"Ann.Univ.Mariae Curie-Sklodoska Sect.A. 55. 115-138 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] D.Partyka, K.Sakan: "On Heinz's inequality"Bull.Soc.Sci.Lettres Lodz. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Shimomura: "On transformations which preserve poly-temperatures of degreem"Mathematical Journal of Ibaraki University. 33. 23-34 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] N.Suzuki, N.A.Watson: "A characterization of heat bolls by mean value property for temperatures"Proc.Amer.Math.Soc.. 129. 2709-2713 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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