Quasiconformal deformation of a ball (Q1191195)

According to Yu. G. Reshetnyak, for every \((1+\varepsilon)\)- quasiconformal mapping \(f\) of the unit ball \(B=B(0,1)\subset{\mathbf R}^ n\), \(n\geq 3\), onto itself normalized by the condition \(f(0)=0\), there exist an orthogonal transformation \(A\) and a continuous vector field \(u(x)\in W^ 1_ p(B)\), \(p>n\), such that \(u(0)=0\), \((x,u(x))|_{\partial B}=0\), \(Af(x)=x+\varepsilon u(x)+o(\varepsilon)\) and \((Qu)_{ij}:={1\over 2} (\partial u_ i/\partial x_ j)+(\partial u_ j/\partial x_ i)-n^{- 1}\delta_{ij}\text{div }u(x)\in L_ \infty(B)\), \(i,j=1,\dots,n\). Using this result, the author establishes compatibility conditions for the system \((Qu)_{ij}=q_{ij}\), \(i,j=1,\dots,n\), with coefficients \(q_{ij}\in L_ \infty(B)\), gives a precise integral representation for its solution \(u(x)\) and derives a variational formula for a \((1+\varepsilon)\)-quasiconformal mapping of \(B\) onto itself, which generalizes a well-known formula of P. P. Belinskij in dimension two.