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 中的边值问题 边界上 Besov 空间上的算子 K 相对于 Besov 范数是有界的 我们可以证明算子从 δD 到 δD 的有界性。 (1) 我们使用扩展算子 E 将 δD 上定义的函数扩展为 R^n。(2) f 的贝索夫范数由 (∫_D |▽f(x)|^<Pλ>dx)^<1/P>，其中 δ(x) 是从 x 到 δD 的距离。 (3) 我们证明算子 F 的有界性，而不是 K 的有界性利用D与D外的最大函数，从D到D外的适当范数。我们证明了算子K的有界性，这对于利用双层势求解Dirichlet问题具有重要意义。

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: "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。

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。