Potential the Oretic study on quasi-regular mappings

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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.