The Extremal Problems For Mappings Of Finite Distortion

The extremal quasiconformal mapping theory is important in the theory of Te-ichmuller spaces.Recently,this theory has been extended to the setting of mappings of finite distortion.In this thesis,we mainly study weighted Grotzsch problem for mappings of finite distortion and the unique extremality of spiral-stretch,mean-while,we also discuss the ?-Nitsche conjecture.This thesis mainly contains four chapters.In chapter 1,we introduce the definition,notations as well as the basic theory of quasiconformal mappings and mappings of finite distortion.Moreover,we introduce historical background of the problems discussed in the thesis and state the main results we have got.In chapter 2,we consider the following extremal problem(?)where F denotes the set of all homeomorphisms f with finite linear distortion K(z,f)between two rectangles Q1 =[0,l]?[0,1]and Q2 =[0,L]×[0,1]taking vertices into vertices,? is strictly convex increasing and ?(x)is a positive weighted function.When ?' is unbounded,the extremal problem has a unique extremal mapping of the form x + iy ?u(x)+ iy.As ?' is bounded,we obtain that when L<l there also exists a uniquely extremal mapping,and when L>l there maybe no solution for minimization problem,but by the method of Martin and McKubre-Jordens we construct a minimizing sequence which attains the minimization in the limit.In this chapter,a similar problem of Nitsche-type,which concerns the minimiser of some weighted functional for mappings of finite distortion between two annuli,is also discussed.In chapter 3,Balogh-Fassler-Platis proved that a spiral-stretch map is extremal for the extremal problem(?)among the set of all Wloc 1/2-homemorpnisrm g with finite linear distortion K(z,f)between two annuli and having same boundary values with the spiral-stretch map.In this chapter,we will give a very fast approach to this result without the g E Wloc 1,2-assurrmption.Furthermore,the unique extremality of spiral-stretch is also obtained.In chapter 4,on the basis of the classical Nitsche conjecture,Kalaj proposed the so-called ?-Nitsche conjecture as follows:There exists a p-harmonic homeomor-phism between two annuli if and only if ?-Nitsche condition holds,In this chapter,we give a positive answer to Kalaj's conjecture for ?(w)=|w|-2.