Potential the Oretic study on quasi-regular mappings

本研究的目的是研究拟共形映射的边界行为，并将我们的方法应用于非线性椭圆型方程的Dirichlet问题，为了完成本研究，我们召集了一些研究人员进行讨论。在科学研究补助金的帮助下，我们成功地在Suri-kaiseki-kenkyusyo，京都Sangyo-University，岐阜大学和广岛大学举办了势能理论会议。特别是在广岛大学，我们与韩国著名数学家共同举办了真实的复分析研讨会，其论文集即将出版，将作为本研究的进一步发展。本研究的首席研究员被邀请到国际会议上发表演讲。通过与国际上许多著名数学家的讨论，我们在这方面的研究中取得了很大的进展。研究一般情况下的单调函数是非常有用的，我们得到了单位球上单调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.