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Potential theory in a domain with fractal boundary

Research Project

Project/Area Number 11640154
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

WATANABE Hisako  Ochanomizu Univ., Graduate School of Humanities and Sciences, Professor, 大学院・人間文化研究科, 教授 (70017193)

Co-Investigator(Kenkyū-buntansha) MAEDA Michie  Ochanomizu University, Faculty of Sciences, Professor, 理学部, 教授 (30017206)
MATSUZAKI Katsuhiko  Ochanomizu University, Faculty of Sciences, Assistant Professor, 理学部, 助教授 (80222298)
TAKEO Fukiko  Ochanomizu University, Faculty of Sciences, Professor, 理学部, 教授 (40109228)
YOSHIDA Hidenobu  Chiba Univ. Graduate School of Natural Sciences, Professor, 大学院・自然科学研究科, 教授 (60009280)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2001: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2000: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordsfractal boundary / double layer potentials / Whitney decomposition / Besov spaces / Besov norms / maximal functions / uniform domains / boundedness of operators / フラクタルな側面 / 熱2重層ポテンシャル / 放物型距離 / 放物型極大関数 / 放物型シリンダー / Besov空間 / Besovノルム / 2重層ポテンシャル / Hausdorff次元 / 特異積分 / Whitney分解 / Hausdorff測度 / The Littlewoods-Paley theory / 非接
Research Abstract

We consider the boundary-value problems in a domain D with fractal boundary. It often occurs that an operator K on the Besov space on the boundary is bounded with respect to the Besov norms. We can prove the boundedness of an operator from δD to δD in the following method.
(1) We extend a function defined on δD to R^n by using an extension operator E.
(2) The Besov norm of f is estimated by (∫_D |▽f(x)|^<Pλ>dx)^<1/P>, where δ(x) is the distance from x to δD.
(3) Instead of the boundedness of K we prove the boundedness of an operator F from D to the outside of D with respect to suitable norms by using the maximal functions between D and the outside of D.
We proved the boundedness of an operator K, which is important to solve the Dirichlet problem by using double layer potentials.

Report

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

    (33 results)

All Other

All Publications (33 results)

  • [Publications] H.Watanabe: "Besov spaces on fractal sets"Josai Math. Monographs. 1. 121-134 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Watanabe: "Boundary behavior of double layer potentials with fractal boundary"Natur Sci. Rep. Ochanomizu Univ.. 50,2. 1-10 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Watanabe: "Uniqueness of double layer potentials for a domain with fractal boundary"Hiroshima Math. J.. 30. 55-77 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Watanabe: "Estimates of the Besov norms on fractal boundary by volume integrals"Natur. Sci. Rep. Ochanomizu Univ. 51,1. 1-10 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Yoshida, I.Miyamoto: "Harmonic functions in a cone which vanish on the boundary"Math. Nachr. 202. 177-187 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Yoshida, I.Miyamoto: "Harmonic functions in a cylinder with the normal derivatives vanishing on the boundary"Ann. olon. Math.. 74. 229-235 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Watanabe: "Besov spaces on fractal sets"Josai Math. Monographs. 1. 121-134 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Watanabe: "boundedness of operators on Besov spaces on a fractal set"数理解析研究講究録1116. 165-180 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Watanabe: "Boundary behavior of double layer potentials with fractal boundary"Natur Sci. Rep. Ochanomizu Univ.. 50,2. 1-10 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Watanabe: "Uniqueness of double layer potentials for a domain with fractal boundary"Hiroshima Math. J.. 30. 55-77 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Watanabe: "Estimates of the Besov norms on fractal boundary by volume integrals"Natur. Sci. Rep. Ochanomizu Univ.. 51. 1 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Watanabe: "Estimates of the Besov norms on fractal lateral boundary by volume integrals"Natur. Sci. Rep. Ochanomizu Univ.. 52(to appear). 1 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Yoshida and I. Miyamoto: "Harmonic functions in a cone which vanish on the boundary"Math. Nachr.. 202. 177-187 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Yoshida and I. Miyamoto: "Harmonic functions in a cylinder with the normal derivatives vanishing on the boundary"R.I.M. Kokyuroku. 1116. 34-44 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Yoshida and I. Miyamoto: "Harmonic functions in a cylinder with the normal derivatives vanishing on the boundary"Ann. Polon. Math.. 74. 229-2235 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Yoshida and I. Miyamoto: "Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone"J. Math. Soc. Japan. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Yoshida, I. Miyamoto and M. Yanagisita: "Beuring-Dahlberg-Sj\"{o}gren type theorems for minimally thin sets in a cone"Canad. Math. Bull. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Watanabe: "Estimates of the Besov norms on fractal lateral boundary by volume integrals"Natur. Sci. Rep. Ochanomizu Univ.. 52,(2). 9-22 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Yoshida, I.Miyamoto: "Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone"J. Math. Soc. Japan. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Yoshida, I.Miyamoto M.Yanagisita: "Beuring-Dahlberg-Sj {o}gren type theorems for minimally thin sets in a cone"Canad. Math. Bull.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Matsui, F.Takeo: "Chaotic semigroups generated by certain differential operators of order 1"SUT Journal of Mathematics. 37. 51-67 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Matsuzaki: "Simply connected domains on a hyperbolic surface"New Zealand J. Math.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Watanabe: "Uniqueness of double layer potentials for a domain with fractal Boundary"Hiroshima Math.J.. 30. 55-77 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Watanabe: "Estimates of the Besov norms on fractal boundary by volume integrals"Natur.Sci.Rep.Ochanomizu Uni.. 51. 1-10 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Yosida and I.Miyamoto: "Harmonic functions in a cylinder with normal derivatives Vanishing on the boundary"Ann.Pol.Math.. 74. 229-235 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Matsui and F.Takeo: "Chaotic semigroups generated by certain differential operators of order 1"第9回関数空間セミナー報告集. 102-107 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Harai and M.Maeda: "Measurable norms and related conditions in some examples"第9回関数空間セミナー報告集. 34-39 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Watanabe: "Besov spaces on fractal sets"Josai Math. Monographs 1. 1. 121-134 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Watanabe: "Boundedness of operators on Besov spaces on a fractal set"数理解析研究所講究録. 1116. 165-180 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Watanabe: "Boundary behavior of double layer potentials with fractal boundary"Natur. Sci. Rep. Ochanomizu Univ.. 50・2. 1-10 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Watanabe: "Uniqueness of double layer potentials for a domain with fractal boundary"Hiroshima Math. J.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Yoshida and I.Miyamoto: "Harmonic functions in a cone which vanishes on the boundary"Math. Nachr.. 202. 177-187 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Matsuzaki: "The Hausdorff dimension of the limit sets of infinitely generated Kleinian groups"Math. Proc. Camb. Phil. Soc.. 129. 123-139 (2000)

    • Related Report
      1999 Annual Research Report

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

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