Potential theory in a domain with fractal boundary

分形边界域中的势理论

• 批准号: 11640154
• 负责人: WATANABE Hisako
• 金额: $ 1.92万
• 依托单位: Ochanomizu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640154/
• 关键词: fractal 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; Hausdo

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.

我们考虑具有分形边界的域D中的边界值问题。经常发生，边界上BESOV空间上的操作员K相对于BESOV规范是绑定的。我们可以在以下方法中证明操作员从ΔD到ΔD的界限。（1）我们通过使用扩展运算符E扩展了在ΔD上定义的功能。通过使用D和D之间的最大功能，我们证明了操作员F从D到D的外部的界限。We提供了操作员K的界限，这对于通过使用双层电势来解决Dirichlet问题很重要。

项目成果

H.Watanabe: "Estimates of the Besov norms on fractal boundary by volume integrals"Natur.Sci.Rep.Ochanomizu Uni.. 51. 1-10 (2000)

H.Watanabe：“通过体积积分对分形边界的贝索夫范数的估计”Natur.Sci.Rep.Ochanomizu Uni.. 51. 1-10 (2000)

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)

H. Watanabe：“分形集上 Besov 空间上的算子有界” 数学分析研究 Kokyuroku 1116. 165-180 (1999)

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)

H. Yoshida 和 I. Miyamoto：“圆柱体中的调和函数，其正常导数在边界上消失”Ann。

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).

H.Yoshida，I.Miyamoto：“锥体中最小薄集和稀疏集的维纳类型的两个准则”J。

H.Watanabe: "Uniqueness of double layer potentials for a domain with fractal boundary"Hiroshima Math. J.. 30. 55-77 (2000)

H.Watanabe：“具有分形边界的域的双层势的唯一性”广岛数学。