This note deals with the existence and uniqueness of a minimiser of the following Grtzsch-type problem inf f ∈F∫∫_(Q_1)φ(K(z,f))λ(x)dxdyunder some mild conditions,where F denotes the set of all homeomorphims f wi...This note deals with the existence and uniqueness of a minimiser of the following Grtzsch-type problem inf f ∈F∫∫_(Q_1)φ(K(z,f))λ(x)dxdyunder some mild conditions,where F denotes the set of all homeomorphims f with finite linear distortion K(z,f)between two rectangles Q_1 and Q_2 taking vertices into vertices,φ is a positive,increasing and convex function,and λ is a positive weight function.A similar problem of Nitsche-type,which concerns the minimiser of some weighted functional for mappings between two annuli,is also discussed.As by-products,our discussion gives a unified approach to some known results in the literature concerning the weighted Grtzsch and Nitsche problems.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11371268 and 11171080)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20123201110002)the Natural Science Foundation of Jiangsu Province(Grant No.BK20141189)
文摘This note deals with the existence and uniqueness of a minimiser of the following Grtzsch-type problem inf f ∈F∫∫_(Q_1)φ(K(z,f))λ(x)dxdyunder some mild conditions,where F denotes the set of all homeomorphims f with finite linear distortion K(z,f)between two rectangles Q_1 and Q_2 taking vertices into vertices,φ is a positive,increasing and convex function,and λ is a positive weight function.A similar problem of Nitsche-type,which concerns the minimiser of some weighted functional for mappings between two annuli,is also discussed.As by-products,our discussion gives a unified approach to some known results in the literature concerning the weighted Grtzsch and Nitsche problems.
