2002 年度実績報告書

関数方程式に付随する複雑集合の総合解析

研究課題

研究課題/領域番号	12440040
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	水田義弘広島大学, 総合科学部, 教授 (00093815) 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) 原惟行大阪府立大学, 工学部, 教授 (20029565) 村田實東京工業大学, 理学部, 教授 (50087079)
キーワード	Denjoy領域 / Martin境界 / 調和関数 / John領域 / 凸集合 / 一様領域 / 境界Harnack原理 / 容量密度条件
研究概要	・Denjoy領域とはその境界が一つの超平面内にのみある領域をいうが,境界が超平面内にあるという条件を緩めて,グラフで挟まれた領域に含まれるときDenjoy型領域と呼ぼう.Denjoy型領域のMartin境界はDenjoy領域に比べて,はるかに複雑になってしまうが,有限位数の正調和関数に限ればDenjoy領域と類似した結果が成立することを示した. ・John領域の各境界点上の極小Martin境界点は有限であり,John定数に応じた定数で押さえられること.またJohn定数が1に十分近いJohn領域に対しては極小Martin境界点の数は、2個以下であることを示した.さらに凸集合の和で表される領域に対して極小Martin境界点の数が1個である条件を与えた. ・境界が容量密度条件を満たすという仮定の基で,John領域や一様領域,一様John領域の特徴付けを行った.John領域のための必要十分条件は調和測度の下からの評価が成立する事であり,一様領域のための必要十分条件はそれに加えて一様な境界Harnack原理が成立することであることを示した.また一様John領域は対応する一様境界Harnack原理で特徴づけられることを示した.

研究成果
(24件)

すべてその他

すべて文献書誌 (24件)

[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] J.Sugie: "On the position of unbounded separatrices for van der Pol's system"Dynam. Systems Appl.. 11. 53-63 (2002)
[文献書誌] J.Sugie, K.Kita, N.Yamaoka: "Oscillation constant of second-order non-linear self-adjoint differential equations"Ann. Mat. Pura Appl.(4). 181. 309-337 (2002)
[文献書誌] J.Sugie, N.Yamaoka: "An infinite sequence of nonoscillation theorems for second-order nonlinea differential equations of Euler type"Nonlinear Anal.. 50. 373-388 (2002)
[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.. 131. 501-509 (2003)
[文献書誌] J.Sugie: "Oscillation criteria of Kneser-Hille type for second-order differential equations with nonlinear perturbed terms"Rocky Mountain J. Math.. (to appear). (2003)
[文献書誌] T.A.Burton, T.Furumochi: "Krasnoselskii's fixed point theorem and stability"Nonlinear Anal.. 49. 445-454 (2002)
[文献書誌] T.A.Burton, T.Furumochi: "Asymptotic behavior of solutions of functional differential equations by fixed point theorems"Dynamic Systems and Applications. 11. 499-519 (2002)
[文献書誌] Y.Saito, T.Hara, W.Ma: "Harmless delays for permanence and impersistence of a Lotka-Volterra discrete predator-prey system"Nonlinear Anal.. 50. 703-715 (2002)
[文献書誌] W.Ma, Y.Takeuchi, T.Hara, et al.: "Permanence of an SIR epidemic model with distributed time delays"Tohoku Math. J.. 54. 581-591 (2002)
[文献書誌] B.Gustafsson, M.Sakai: "Sharp estimates of the curvature of some free boundaries in two dimensions"Ann. Acad. Sci. Fenn.. (to appear). (2003)
[文献書誌] M.Jeong, M.Taniguchi: "Bell representations of finitely connected planar domains"Proc. AMS. (to appear). (2003)
[文献書誌] M.Taniguchi: "Synthetic deformation space of an entire function"Contemporary Math.. 303. 107-136 (2002)
[文献書誌] M.Taniguchi: "Size of the Julia set of a structurally finite transcendental entire function"Math. Proc. Camb. Phil. Soc.. (to appear). (2003)
[文献書誌] Y.Mizuta, T.Shimomura: "Holder continuity and differentiability for weighted Sobolev spaces"Proc. Amer. Math. Soc.. 130. 2985-2994 (2002)
[文献書誌] T.Futamura, K.Kishi, Y.Mizuta: "Removability of sets for sub-polyharmonic functions"Hiroshima Math J.. (to appear). (2003)
[文献書誌] T.Futamura, Y.Mizuta: "Tangential limits and removable sets for weighted Sobolev spaces"Hiroshima Math J.. (to appear). (2003)
[文献書誌] M.Murata: "Martin boundaries of elliptic skew products, semismall perturbations, an foundamental solutions of parabolic equations"J. Funct. Anal.. 194. 53-141 (2002)
[文献書誌] M.Murata: "Heat escape"Math. Ann.. (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)

2002 年度 実績報告書

関数方程式に付随する複雑集合の総合解析

研究代表者

相川 弘明 島根大学, 総合理工学部, 教授 (20137889)

研究成果

[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] J.Sugie: "On the position of unbounded separatrices for van der Pol's system"Dynam. Systems Appl.. 11. 53-63 (2002)

[文献書誌] J.Sugie, K.Kita, N.Yamaoka: "Oscillation constant of second-order non-linear self-adjoint differential equations"Ann. Mat. Pura Appl.(4). 181. 309-337 (2002)

[文献書誌] J.Sugie, N.Yamaoka: "An infinite sequence of nonoscillation theorems for second-order nonlinea differential equations of Euler type"Nonlinear Anal.. 50. 373-388 (2002)

[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.. 131. 501-509 (2003)

[文献書誌] J.Sugie: "Oscillation criteria of Kneser-Hille type for second-order differential equations with nonlinear perturbed terms"Rocky Mountain J. Math.. (to appear). (2003)

[文献書誌] T.A.Burton, T.Furumochi: "Krasnoselskii's fixed point theorem and stability"Nonlinear Anal.. 49. 445-454 (2002)

[文献書誌] T.A.Burton, T.Furumochi: "Asymptotic behavior of solutions of functional differential equations by fixed point theorems"Dynamic Systems and Applications. 11. 499-519 (2002)

[文献書誌] Y.Saito, T.Hara, W.Ma: "Harmless delays for permanence and impersistence of a Lotka-Volterra discrete predator-prey system"Nonlinear Anal.. 50. 703-715 (2002)

[文献書誌] W.Ma, Y.Takeuchi, T.Hara, et al.: "Permanence of an SIR epidemic model with distributed time delays"Tohoku Math. J.. 54. 581-591 (2002)

[文献書誌] B.Gustafsson, M.Sakai: "Sharp estimates of the curvature of some free boundaries in two dimensions"Ann. Acad. Sci. Fenn.. (to appear). (2003)

[文献書誌] M.Jeong, M.Taniguchi: "Bell representations of finitely connected planar domains"Proc. AMS. (to appear). (2003)

[文献書誌] M.Taniguchi: "Synthetic deformation space of an entire function"Contemporary Math.. 303. 107-136 (2002)

[文献書誌] M.Taniguchi: "Size of the Julia set of a structurally finite transcendental entire function"Math. Proc. Camb. Phil. Soc.. (to appear). (2003)

[文献書誌] Y.Mizuta, T.Shimomura: "Holder continuity and differentiability for weighted Sobolev spaces"Proc. Amer. Math. Soc.. 130. 2985-2994 (2002)

[文献書誌] T.Futamura, K.Kishi, Y.Mizuta: "Removability of sets for sub-polyharmonic functions"Hiroshima Math J.. (to appear). (2003)

[文献書誌] T.Futamura, Y.Mizuta: "Tangential limits and removable sets for weighted Sobolev spaces"Hiroshima Math J.. (to appear). (2003)

[文献書誌] M.Murata: "Martin boundaries of elliptic skew products, semismall perturbations, an foundamental solutions of parabolic equations"J. Funct. Anal.. 194. 53-141 (2002)

[文献書誌] M.Murata: "Heat escape"Math. Ann.. (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)

2002 年度実績報告書

相川弘明島根大学, 総合理工学部, 教授 (20137889)