2003 Fiscal Year Annual Research Report

ポテンシャル解析

Research Project

Project/Area Number	15340046
Research Category	Grant-in-Aid for Scientific Research (B)
Research Institution	Shimane University
Principal Investigator	相川弘明島根大学, 総合理工学部, 教授 (20137889)
Co-Investigator(Kenkyū-buntansha)	水田義弘広島大学, 総合科学部, 教授 (00093815) 町原秀二島根大学, 総合理工学部, 助手 (20346373) 黒岩大史島根大学, 総合理工学部, 助教授 (40284020) 鈴木紀明名古屋大学, 大学院・多元数理科研究科, 助教授 (50154563)
Keywords	容量密度条件 / 内部一様領域 / 調和関数 / Fatou定理 / Carleson評価 / Dirac方程式 / ソボレフ関数 / 熱方程式
Research Abstract	・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. ・非線型Dirac方程式の初期値問題の可解性を調べた.初期値になるべく滑らかさを課さない条件の下で考察した.今年度は特に空間1次元における調査を主とした. ・ソボレフ関数の境界値の存在について研究を行った。一般のソボレフ関数に関しては細極限値の存在を,monotoneソボレフ関数に関しては非接極限値の存在を論じた。・熱方程式の解が平均値の性質を満たす密度関数の存在と非存在,有界な密度関数の存在などについて解析した.α-放物型作用素の定めるBergman空間の基本的性質を調べた.

Research Products
(19 results)

All Other

All Publications (19 results)

[Publications] H.Aikawa, T.Luhdh, T.Mizutani: "Martin boundary of a fractal domain"Potential Anal.. 18. 311-357 (2003)
[Publications] 相川弘明: "複雑領域のMartin境界と境界Harnack原理"数学. 55. 1-19 (2003)
[Publications] H.Aikawa: "Positive harmonic functions of finite order in a Denjoy type domain"Proc.Amer.Math.Soc.. 131. 3873-3881 (2003)
[Publications] H.Aikawa: "Martin boundary and boundary harnack principle for non-smooth domains"Sugaku Expositions. (to appear). (2004)
[Publications] H.Aikawa: "Potential theoretic characterizations of nonsmooth domains"Bull.London Math.Soc.. (to appear). (2004)
[Publications] H.Aikawa: "Fatou and Littlewood theorems for Poisson integrals with respect to non-integrabla kernels"Complex Variables Theory Appl.. (to appear). (2004)
[Publications] S.Machihara, K.Nakanishi, T.Ozawa: "Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation"Revista Matematica Iberoamericana. 19. 179-194 (2003)
[Publications] S.Machihara, M.Nakamura, T.Ozawa: "Small global solutions for nonlinear Dirac equation"Differential and Integral Equation. (to appear). (2004)
[Publications] D.Kuroiwa: "Existence theorems of set optimization with set-valued maps"Journal of Information & Optimization Sciences. 24. 73-84 (2003)
[Publications] S.Matsushita, D.Kuroiwa: "Approximation fixed points of nonexpansive nonself-mappings"Scientiae Mathematicae Japonicae. 57. 171-176 (2003)
[Publications] D.Kuroiwa: "Set optimization with weighted criteria and efficiency"Proceedings of the Second International Conference on Nonlinear Analysis and Convex Analysis. 233-238 (2003)
[Publications] S.Matsushita, D.Kuroiwa: "Some observations of approximation of fixed points of nonexpansive nonself-mappings"Proceedings of the Second International Conference on Nonlinear Analysis and Convex Analysis. 275-280 (2003)
[Publications] D.Kuroiwa: "Existence of efficient points of set optimization with weighted criteria"Journal of Nonlinear and Convex Analysis. 4. 117-124 (2003)
[Publications] S.Matsushita, D.Kuroiwa: "Strong convergence of averaging iterations of nonexpansive nonself-mappings"Journal of Mathematical Analysis and Applications. (to appear). (2004)
[Publications] Y.Mizuta, T.Shimomura: "Minimally fine limits at infinity for p-precise functions"J.Math.Soc.Japan. 56. 1-15 (2004)
[Publications] T.Futamura, Y.Mizuta: "Lindelof theorems for monotone Sobolev functions"Acad.Sci.Fenn.Ser.A.I.Math.. 28. 271-227 (2003)
[Publications] N.Suzuki, N.A.Watson: "Mean value densities for temperatures"Colloquium Mathematicum. 98. 87-96 (2003)
[Publications] K.Shimomura, N.Suzuki, M.Nishio: "α-parabolic Bergman spaces and their reproducing kernels"数理解析研究所講究録. 1352. 106-113 (2004)
[Publications] 相川弘明(分担執筆): "数学辞典第4版の4項目(調和関数,優(劣)調和関数,ディリクレ問題,ポテンシャル論,容量)"日本数学会(to appear). (2004)

2003 Fiscal Year Annual Research Report

ポテンシャル解析

Principal Investigator

相川 弘明 島根大学, 総合理工学部, 教授 (20137889)

Research Products

[Publications] H.Aikawa, T.Luhdh, T.Mizutani: "Martin boundary of a fractal domain"Potential Anal.. 18. 311-357 (2003)

[Publications] 相川弘明: "複雑領域のMartin境界と境界Harnack原理"数学. 55. 1-19 (2003)

[Publications] H.Aikawa: "Positive harmonic functions of finite order in a Denjoy type domain"Proc.Amer.Math.Soc.. 131. 3873-3881 (2003)

[Publications] H.Aikawa: "Martin boundary and boundary harnack principle for non-smooth domains"Sugaku Expositions. (to appear). (2004)

[Publications] H.Aikawa: "Potential theoretic characterizations of nonsmooth domains"Bull.London Math.Soc.. (to appear). (2004)

[Publications] H.Aikawa: "Fatou and Littlewood theorems for Poisson integrals with respect to non-integrabla kernels"Complex Variables Theory Appl.. (to appear). (2004)

[Publications] S.Machihara, K.Nakanishi, T.Ozawa: "Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation"Revista Matematica Iberoamericana. 19. 179-194 (2003)

[Publications] S.Machihara, M.Nakamura, T.Ozawa: "Small global solutions for nonlinear Dirac equation"Differential and Integral Equation. (to appear). (2004)

[Publications] D.Kuroiwa: "Existence theorems of set optimization with set-valued maps"Journal of Information & Optimization Sciences. 24. 73-84 (2003)

[Publications] S.Matsushita, D.Kuroiwa: "Approximation fixed points of nonexpansive nonself-mappings"Scientiae Mathematicae Japonicae. 57. 171-176 (2003)

[Publications] D.Kuroiwa: "Set optimization with weighted criteria and efficiency"Proceedings of the Second International Conference on Nonlinear Analysis and Convex Analysis. 233-238 (2003)

[Publications] S.Matsushita, D.Kuroiwa: "Some observations of approximation of fixed points of nonexpansive nonself-mappings"Proceedings of the Second International Conference on Nonlinear Analysis and Convex Analysis. 275-280 (2003)

[Publications] D.Kuroiwa: "Existence of efficient points of set optimization with weighted criteria"Journal of Nonlinear and Convex Analysis. 4. 117-124 (2003)

[Publications] S.Matsushita, D.Kuroiwa: "Strong convergence of averaging iterations of nonexpansive nonself-mappings"Journal of Mathematical Analysis and Applications. (to appear). (2004)

[Publications] Y.Mizuta, T.Shimomura: "Minimally fine limits at infinity for p-precise functions"J.Math.Soc.Japan. 56. 1-15 (2004)

[Publications] T.Futamura, Y.Mizuta: "Lindelof theorems for monotone Sobolev functions"Acad.Sci.Fenn.Ser.A.I.Math.. 28. 271-227 (2003)

[Publications] N.Suzuki, N.A.Watson: "Mean value densities for temperatures"Colloquium Mathematicum. 98. 87-96 (2003)

[Publications] K.Shimomura, N.Suzuki, M.Nishio: "α-parabolic Bergman spaces and their reproducing kernels"数理解析研究所講究録. 1352. 106-113 (2004)

[Publications] 相川弘明(分担執筆): "数学辞典第4版の4項目(調和関数,優(劣)調和関数,ディリクレ問題,ポテンシャル論,容量)"日本数学会(to appear). (2004)

相川弘明島根大学, 総合理工学部, 教授 (20137889)