Research on potential problems from various aspects

2010 Fiscal Year Self-evaluation Report

• Project/Area Number: 20244007
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Hokkaido University
• Principal Investigator: AIKAWA Hiroaki Hokkaido University, 大学院・理学研究院, 教授 (20137889)
• Project Period (FY): 2008 – 2012
• Keywords: ポテンシャル / 調和

Research Abstract

(1)古典的ポテンシャル論は調和,優調和,劣調和,多重劣調和関数などの基本関数の定性,定量的な性質を調べてきた.これらの基本関数に関する問題をポテンシャル問題と捉え,多様な方向から多面的に研究する
(2)具体的にはポテンシャル問題を以下のように分類し,研究分担者・連携研究者・研究協力者が分担しあって研究を遂行する
1.複雑領域におけるポテンシャル問題
2.フラクタルにおけるポテンシャル問題
3.多様体におけるポテンシャル問題
4.関数空間におけるポテンシャル問題
5.ポテンシャル問題の応用

Research Products

• [Journal Article] Modulus of continuity of the Dirichlet solutions (2010)
  • Author(s): H. Aikawa
  • Journal Title: Bull. London Math. Soc. 42 Pages: 857-867
  • Peer Reviewed
• [Journal Article] On the existence of mean curvature flow with transport term (2010)
  • Author(s): C. Liu, N. Sato, Y. Tonegawa
  • Journal Title: Interfaces and Free Boundaries 12 Pages: 251-277
  • Peer Reviewed
• [Journal Article] Weighted Orlicz-Riesz capacity of balls (2010)
  • Author(s): Y. Mizuta, T. Ohno, T. Shimomura
  • Journal Title: Proc. Amer. Math. Soc. 138 Pages: 4291-4302
  • Peer Reviewed
• [Journal Article] Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees (2010)
  • Author(s): J. Kigami
  • Journal Title: Adv. Math. 225 Pages: 2674-2730
  • Peer Reviewed
• [Journal Article] Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations (2009)
  • Author(s): M. Murata
  • Journal Title: J. Func. Anal. 257 Pages: 1799-1827
  • Peer Reviewed
• [Journal Article] Riemann mappings of invariant components of Kleinian groups (2009)
  • Author(s): H. Shiga
  • Journal Title: J. London Math. Soc. 80 Pages: 716-728
  • Peer Reviewed
• [Journal Article] Parabolic, Harnack inequality and heat kernel estimates for random walks with long range jumps (2009)
  • Author(s): M.T. Barlow, R.F. Bass, T. Kumagai
  • Journal Title: Math. Z. 261 Pages: 297-320
  • Peer Reviewed
• [Journal Article] Equivalence between the boundary Harnack principle and the Carleson estimate (2008)
  • Author(s): H. Aikawa
  • Journal Title: Math. Scand. 103 Pages: 61-76
  • Peer Reviewed
• [Journal Article] Doubling conditions for harmonic measure in John domains (2008)
  • Author(s): H. Aikawa, K. Hirata
  • Journal Title: Ann. Inst. Fourier (Grenoble) 58(2) Pages: 429-445
  • Peer Reviewed
• [Journal Article] The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations (2008)
  • Author(s): K. Hirata
  • Journal Title: Math. Ann. 340 Pages: 625-645
  • Peer Reviewed
• [Presentation] Quaiconformal modification of metrics on self-similar sets (2010)
  • Author(s): J. Kigami
  • Organizer: 1062nd AMS meeting, Analysis, Probability and Mathematical Physics on Fractals
  • Place of Presentation: Syracuse University, USA
  • Year and Date: 20101002-20101003
• [Presentation] Convergence of mixing times for sequences of simple random walks on graphs (2010)
  • Author(s): T. Kumagai
  • Organizer: Combinatorics and Analysis in Spatial Probability
  • Place of Presentation: Eurandom, Eindhoven, the Netherlands
  • Year and Date: 2010-12-13
• [Presentation] Teichmuller spaces and holomorphic maps, Teichmuller theory (2010)
  • Author(s): H. Shiga
  • Place of Presentation: MFO, Oberwolfach, Germany
  • Year and Date: 2010-12-01
• [Presentation] Boundary Harnack principle and Carleson estimate (2010)
  • Author(s): H. Aikawa
  • Organizer: 4th International Conference on Stochastic Analysis and Its applications
  • Place of Presentation: 関西大学
  • Year and Date: 2010-08-31
• [Presentation] 複雑領域上のポテンシャル論― 解析的性質と幾何的性質― (2010)
  • Author(s): 相川弘明
  • Organizer: 日本数学会
  • Place of Presentation: 慶應義塾大学
  • Year and Date: 2010-03-24
• [Presentation] Existence of weak mean curvature flow with non-smooth forcing term (2009)
  • Author(s): Y. Tonegawa
  • Organizer: SIAM conference on Analysis of PDEs
  • Place of Presentation: Miami, U.S.A.
  • Year and Date: 2009-12-07
• [Presentation] 非線形不等式を満たす優調和関数の境界挙動 (2009)
  • Author(s): 平田賢太郎
  • Organizer: 日本数学会
  • Place of Presentation: 大阪大学(特別講演)
  • Year and Date: 2009-09-26
• [Presentation] John domains and the doubling property of the harmonic measure (2009)
  • Author(s): H. Aikawa
  • Organizer: Summer School in Conformal Geometry, Potential Theory, and Applications
  • Place of Presentation: NUI Maynooth, Ireland
  • Year and Date: 2009-06-27
• [Presentation] 複雑領域上の正調和関数 (2009)
  • Author(s): 相川弘明
  • Organizer: 日本数学会
  • Place of Presentation: 東京大学(企画特別講演)
  • Year and Date: 2009-03-29
• [Book] Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets, Memoirs of the AMS/199/932 (2009)
  • Author(s): J.Kigami
  • Total Pages: 94
  • Publisher: アメリカ数学会
• [Book] 複雑領域上のディリクレ問題- ポテンシャル論の観点から(岩波数学叢書) (2008)
  • Author(s): 相川弘明
  • Total Pages: 302
  • Publisher: 岩波書店