Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Hölder continuity of harmonic quasiconformal mappings | Arsenovic et al. Journal of Inequalities and Applications 2011 2011 37 http content 2011 1 37 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Holder continuity of harmonic quasiconformal mappings Milos Arsenovic 1 Vesna Manojlovic2 and Matti Vuorinen3 Correspondence arsenovic@matf. department of Mathematics University of Belgrade Studentski Trg 16 11000 Belgrade Serbia Full list of author information is available at the end of the article Springer Abstract We prove that for harmonic quasiconformal mappings a-Holder continuity on the boundary implies a-Holder continuity of the map itself. Our result holds for the class of uniformly perfect bounded domains in fact we can allow that a portion of the boundary is thin in the sense of capacity. The problem for general bounded domains remains open. MSC 2010 30C65. Keywords quasi-conformal maps harmonic mappings H o lder continuity. 1. Introduction The following theorem is the main result in 1 . Theorem . Let D be a bounded domain in R and let fbe a continuous mapping of D into Rn which is quasiconformal in D. Suppose that for some M 0 and 0 a 1 If x - f y l M x - y whenever x and y lie on dD. Then If x - f y l w - yl for all x and y on D where ft min a K1 1 n and M depends only on M a n K f and diam D . The exponent b is the best possible as the example of a radial quasiconformal map f x x a-1x 0 a 1 of Bn onto itself shows see 2 p. 49 . Also the assumption of boundedness is essential. Indeed one can consider g x lxlax x 1 where a 0. Then g is quasiconformal in D Rn Bn see 2 p. 49 it is identity on dD and hence Lipschitz continuous on dD. However Ig te1 - g e1 I X ta 1 t and therefore g is not globally Lipschitz continuous on D. This paper deals with the following question suggested by P. Koskela is it possible to replace b with a if we assume in addition to quasiconformality that f is harmonic In the special case D Bn .
