複雑集合の内と外のポテンシャル解析

• 研究課題/領域番号: 13874023
• 研究種目: 萌芽研究
• 配分区分: 補助金
• 研究分野: 基礎解析学
• 研究機関: 島根大学
• 研究代表者: 相川 弘明 島根大学, 総合理工学部, 教授 (20137889)
• 研究分担者: 熊谷 隆 京都大学, 数理解析研究所, 助教授 (90234509) ; 杉江 実郎 島根大学, 総合理工学部, 教授 (40196720) ; 山崎 稀嗣 島根大学, 総合理工学部, 教授 (70032935)
• 研究期間 (年度): 2001 – 2003
• 研究課題ステータス: 完了 (2003年度)
• 配分額: 2,000千円 (直接経費: 2,000千円); 2003年度: 700千円 (直接経費: 700千円); 2002年度: 600千円 (直接経費: 600千円); 2001年度: 700千円 (直接経費: 700千円)
• キーワード: 容量密度条件 / 内部一様領域 / 調和関数 / Fatou定理 / Martin境界 / Carleson評価 / Besov空間 / フラクタル / Denjoy領域 / John領域 / 一様領域 / 境界Harnack原理 / Dirichlet解 / 非線形方程式 / Lienard系 / 確率過程 / ベゾフ空間

研究概要

・境界の容量密度条件の下で,一様領域,内部一様領域,John領域が境界Harnack原理や調和測度の評価で特徴付けられることを示した.また,非可積分な核に対する境界挙動を調べ,Fatou型の定理とLittlewood型の定理を導いた.3G不等式を内部一様領域に対して示すとともに,3次元以上の次元ではMartin境界が位相境界と一致するにもかかわらず,Cranston-McConnellの不等式が成立せず,その結果3G不等式が成立しない領域の例を構成した.p-調和関数に関するCarleson評価を導いた.
・ベソフ空間の理論の応用として、フラクタルを典型例として持つd-setと呼ばれるクラスの上に構成した飛耀型対称確率過程が、Triebel氏を初めとした関数空間の専門家が研究している作用素とどのような関係にあるかを調べ、論文にまとめた。

研究成果

• [文献書誌] H.Aikawa, T.Lundh, T.Mizutani: "Martin boundary of a fractal domain"Potential Anal.. 18. 311-357 (2003)
• [文献書誌] 相川弘明: "複雑領域のMartin境界と境界Harnack原理"数学. 55. 1-19 (2003)
• [文献書誌] H.Aikawa: "Positive harmonic functions of finite order in a Denjoy type domain"Proc.Amer.Math.Soc.. 131. 3873-3881 (2003)
• [文献書誌] H.Aikawa: "Martin boundary and boundary harnack principle for non-smooth domains"Sugaku Expositions. (to appear). (2004)
• [文献書誌] H.Aikawa: "Potential theoretic characterizations of nonsmooth domains"Bull.London Math.Soc.. (to appear). (2004)
• [文献書誌] H.Aikawa: "Fatou and Littlewood theorems for Poisson integrals with respect to non-integrable kernels"Complex Variables Theory Appl.. (to appear). (2004)
• [文献書誌] Z.Q.Chen, T.Kumagai: "Heat kernel estimates for stable-like processes on d-sets"Stoch.Proc.Their Appl.. 108. 27-62 (2003)
• [文献書誌] B.M.Hambly, T.Kumagai: "Diffusion processes on fractal fields and their large deviations"Probab.Theory Relat.Fields. 127. 305-352 (2003)
• [文献書誌] B.M.Hambly, T.Kumagai: "Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries"Proc.of Symposia in Pure Math. (to appear). (2004)
• [文献書誌] T.Kumagai: "Homogenization of Finitely Ramified Fractals"Advanced Studies in Pure Math.. 41(to appear). (2004)
• [文献書誌] B.M.Hambly, T.Kumagai: "Heat Kernel Estimates and Law of the Iterated Logarithm for Symmetric Random Walks on Fractal Graphs"Contemporary Mathematics. (to appear). (2004)
• [文献書誌] T.Kumagai: "Function spaces and stochastic processes on fractals"Progr.Probab.,Birkhauser. (to appear). (2004)
• [文献書誌] 相川弘明(分担執筆): "数学事典第4版の4項目(調和関数,優(劣)調和関数,ディレクレ問題,ポテンシャル論,容量)"日本数学会(to appear). (2004)
• [文献書誌] H.Aikawa: "Holder continuity of the Dirichlet solution for a general domain"Bull. London Math. Soc.. 34. 691-702 (2002)
• [文献書誌] H.Aikawa, K.Hirata, T.Lund: "Martin boundary for union of convex sets"数理解析研究所講究録「ポテンシャル論とその周辺」. 1293. 1-14 (2002)
• [文献書誌] 相川弘明: "複雑領域のMartin境界と境界Harnack原理"数学. 55. 1-19 (2003)
• [文献書誌] H.Aikawa, T.Lundh, T.Mizutani: "Martin boundary of a fractal domain"Potential Anal.. (to appear). (2003)
• [文献書誌] H.Aikawa: "Positive harmonic functions of finite order in a Denjoy type domain"Proc. Amer. Math. Soc.. (to appear). (2003)
• [文献書誌] B.M.Hambly, T.Kumagal: "Asymptotics for the spectral and walk dimension as fractals approach Euclidean space"Fractals. 10. 403-412 (2002)
• [文献書誌] R.F.Bass, T.Kumagai: "Laws of the iterated logarithm for the range of random walks in two and three dimensions"Ann. Probab.. 30. 1369-1396 (2002)
• [文献書誌] B.M.Hambly, J.Kigami, T.Kumagai: "Multifractal formalisms for the local spectral and walk dimensions"Math. Proc. Cambridge Philos. Soc.. 132. 555-571 (2002)
• [文献書誌] T.Kumagai: "Function spaces and stochastic processes on fractals"数理解析研究所講究録. 1293. 42-54 (2002)
• [文献書誌] B.M.Hambly, T.Kumagai: "Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries"Proc. of Symposia in Pure Math. (AMS). (to appear). (2003)
• [文献書誌] 相川弘明(編集): "数理解析研究所講究録「ポテンシャル論とその周辺」1293"京都大学数理解析研究所. 198 (2002)
• [文献書誌] 熊谷 隆: "確率論"共立出版. 224 (2003)
• [文献書誌] H.Aikawa: "Holder continuity of Dirichlet solution for a general domain"Bull. London Math. Soc.. (to appear).
• [文献書誌] J.Sugie, K.Kita: "Oscillation criteria for second order nonlinear differential equations of Euler type"J. Math. Anal. Appl.. 253. 414-439 (2001)
• [文献書誌] J.Sugie: "Lienard dynamics with an open limit orbit"NoDEA Nonlinear Differential Equations Appl.. 8. 83-97 (2001)
• [文献書誌] J.Sugie: "A nonoscillation theorem for second order nonlinear differential equations with decaying coefficients"Bull. London Math. Soc.. 33. 299-308 (2001)
• [文献書誌] J.Sugie, N.Yamada, Y.Obata: "Nonoscillation theorems for a nonlinear self-adjoint differential equation"Nonlinear Anal.. 47. 4433-4444 (2001)
• [文献書誌] J.Sugie, N.Yamada: "An infinite sequence of nonoscillation theorems for second order nonlinear differential equations of Euler type"Nonlinear Anal.. (to appear).
• [文献書誌] J.Sugie, K.Kita, N.Yamaoka: "Oscillation constant of second order nonlinear self-adjoint differential equations"Ann. Mat. Pura Appl.. (to appear).
• [文献書誌] J.Sugie, N.Yamaoka: "Applications of phase plane analysis of a Lienard system to positive solutions of Schrodinger equations"Proc. Amer. Math. Soc.. (to appear).
• [文献書誌] J.Sugie, N.Yamaoka: "Nonoscillation theorems for second order nonlinear differential differential equations of Euler type"数理解析研究所講究録. 1216. 224-235 (2001)
• [文献書誌] B.M.Hambly, T.Kumagai: "Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gasket"Stoch. Proc. Appl.. 92. 61-85 (2001)
• [文献書誌] M.T.Barlow, T.Kumagai: "Transition density asymptotics for some diffusion processes with multi-fractal structures"Electronic Journal of Probability. 6. 1-23 (2001)
• [文献書誌] T.Kumagai: "Construction of diffusion processes penetrating fractals -An application of the theory of Besov spaces -"数理解析研究所講究録. 1235. 91-114 (2001)
• [文献書誌] R.F.Bass, T.Kumagai: "Laws of the iterated logarithm for the range of random walks in two and three dimensions"Ann. Probab.. (to appear).
• [文献書誌] B.M.Hambly, J.Kigami, T.Kumagai: "Multifractal formalisms for the local spectral and walk dimensions"Maths. Proc. Cam. Phil. Soc.. (to appear).
• [文献書誌] B.M.Hambly, T.Kumagai: "Asymptotics for the spectral and walk dimension as fractals approach Euclidean space"Fractals. (to appear).
• [文献書誌] T.Kumagai: "Some remarks for jump processes on fractals"Proc. of the Conference held in Graz. (to appear). (2001)