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
Let f ： Ω→ f（Ω） belong to R^n be a W^1,1-homeomorphism with L^1-inegrable inner We show that finiteness of min{lipf（x）, kf（x））, for every x∈ Ω／E, implies that f^-1 ∈ W^1,n and has finite distortion, pro...Let f ： Ω→ f（Ω） belong to R^n be a W^1,1-homeomorphism with L^1-inegrable inner We show that finiteness of min{lipf（x）, kf（x））, for every x∈ Ω／E, implies that f^-1 ∈ W^1,n and has finite distortion, provided that the exceptional set E has σ-finite H^1-measure.Moreover, f has finite distortion, differentiable a.e. and the Jacobian Jf 〉 0 a.e.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.
基金Supported partially by the Academy of Finland(Grant No.131477)the Magnus Ehrnrooth foundation
文摘Let f ： Ω→ f（Ω） belong to R^n be a W^1,1-homeomorphism with L^1-inegrable inner We show that finiteness of min{lipf（x）, kf（x））, for every x∈ Ω／E, implies that f^-1 ∈ W^1,n and has finite distortion, provided that the exceptional set E has σ-finite H^1-measure.Moreover, f has finite distortion, differentiable a.e. and the Jacobian Jf 〉 0 a.e.
