2002 Fiscal Year Annual Research Report

関数方程式に付随する複雑集合の総合解析

Research Project

Project/Area Number	12440040
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) 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)
Keywords	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
Research Abstract	・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.

Research Products
(24 results)

All Other

All Publications (24 results)

[Publications] H.Aikawa: "Holder continuity of the Dirichlet solution for a general domain"Bull. London Math. Soc.. 34. 691-702 (2002)
[Publications] 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)
[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, T.Lundh, T.Mizutani: "Martin boundary of a fractal domain"Potential Anal.. (to appear). (2003)
[Publications] H.Aikawa: "Positive harmonic functions of finite order in a Denjoy type domain"Proc. Amer. Math. Soc.. (to appear). (2003)
[Publications] J.Sugie: "On the position of unbounded separatrices for van der Pol's system"Dynam. Systems Appl.. 11. 53-63 (2002)
[Publications] 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)
[Publications] 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)
[Publications] 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)
[Publications] J.Sugie: "Oscillation criteria of Kneser-Hille type for second-order differential equations with nonlinear perturbed terms"Rocky Mountain J. Math.. (to appear). (2003)
[Publications] T.A.Burton, T.Furumochi: "Krasnoselskii's fixed point theorem and stability"Nonlinear Anal.. 49. 445-454 (2002)
[Publications] 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)
[Publications] 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)
[Publications] 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)
[Publications] B.Gustafsson, M.Sakai: "Sharp estimates of the curvature of some free boundaries in two dimensions"Ann. Acad. Sci. Fenn.. (to appear). (2003)
[Publications] M.Jeong, M.Taniguchi: "Bell representations of finitely connected planar domains"Proc. AMS. (to appear). (2003)
[Publications] M.Taniguchi: "Synthetic deformation space of an entire function"Contemporary Math.. 303. 107-136 (2002)
[Publications] M.Taniguchi: "Size of the Julia set of a structurally finite transcendental entire function"Math. Proc. Camb. Phil. Soc.. (to appear). (2003)
[Publications] Y.Mizuta, T.Shimomura: "Holder continuity and differentiability for weighted Sobolev spaces"Proc. Amer. Math. Soc.. 130. 2985-2994 (2002)
[Publications] T.Futamura, K.Kishi, Y.Mizuta: "Removability of sets for sub-polyharmonic functions"Hiroshima Math J.. (to appear). (2003)
[Publications] T.Futamura, Y.Mizuta: "Tangential limits and removable sets for weighted Sobolev spaces"Hiroshima Math J.. (to appear). (2003)
[Publications] M.Murata: "Martin boundaries of elliptic skew products, semismall perturbations, an foundamental solutions of parabolic equations"J. Funct. Anal.. 194. 53-141 (2002)
[Publications] M.Murata: "Heat escape"Math. Ann.. (to appear). (2003)
[Publications] 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 Fiscal Year Annual Research Report

関数方程式に付随する複雑集合の総合解析

Principal Investigator

相川 弘明 島根大学, 総合理工学部, 教授 (20137889)

Research Products

[Publications] H.Aikawa: "Holder continuity of the Dirichlet solution for a general domain"Bull. London Math. Soc.. 34. 691-702 (2002)

[Publications] 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)

[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, T.Lundh, T.Mizutani: "Martin boundary of a fractal domain"Potential Anal.. (to appear). (2003)

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

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

[Publications] 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)

[Publications] 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)

[Publications] 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)

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

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

[Publications] 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)

[Publications] 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)

[Publications] 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)

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

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

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

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

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

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

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

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

[Publications] M.Murata: "Heat escape"Math. Ann.. (to appear). (2003)

[Publications] 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)

相川弘明島根大学, 総合理工学部, 教授 (20137889)