Potential the Oretic study on quasi-regular mappings

本文研究拟共形映射的边界行为，并将此方法应用于非线性椭圆方程的Dirichlet问题。为了完成目前的研究，我们与研究者召开了一些会议，并相互讨论。在这项科研资助的帮助下，我们成功地在水里怀石料理-健九宫、京都商事大学、岐阜大学和广岛大学举办了势理论会议。特别是在广岛大学，我们与著名的韩国数学家举行了实和复分析研讨会，他们的会议记录将很快出版，并将作为本研究的进一步发展。本研究的首席研究员应邀参加国际会议并作报告。通过与世界上许多著名数学家的讨论，我们成功地收集了这一研究的新方面。拟正则映射的每个坐标都称为单调的。在一般情况下研究单调函数是非常有用的，我们在单位球上得到了单调Sobolev函数的边界行为的有趣结果。我们的研究结果将在一些论文中广泛发表。

The aim of this study is to investigate boundary behaviors of quasi-conformal mappings, and then apply our methods to the Dirichlet problem for non-linear elliptic equations.In order to complete the present study, we called some meetings with investigators and discussed each other. With the help of this Grant-in-aid for Scientific Research, we successfully had the conferences on potential theory in Suuri-kaiseki-kenkyusyo, Kyoto Sangyo-University, Gifu University and Hiroshima University. Especially, in Hiroshima University, we held the workshop on real and complex analysis with famous Korean mathematicians, whose proceedings will be published soon and will be served as the further development of this study.The Head investigator of this study was invited to the international conferences to present and give a talk. We had the great success in collecting new aspects of this study by the discussions with many famous mathematicians in the world.Each coordinate of quasi-regular mappings are called monotone. It is very useful for us to study monotone functions in general settings, We have obtained interesting results on the boundary behaviors for monotone Sobolev functions in the unit ball. Our results will be widely delivered in some papers.