Harmonic maps between two concentric annuli in 𝐑3

Given two annuli {\mathbb{A}(r,R)} and {\mathbb{A}(r_{\ast},R_{\ast})} , in {\mathbf{R}^{3}} equipped with the Euclidean metric and the weighted metric {\lvert y\rvert^{-2}} , respectively, we minimize the Dirichlet integral, i.e., the functional \mathscr{F}[f]=\int_{\mathbb{A}(r,R)}\frac{\lVert Df\rVert^{2}}{\lvert f\rvert% ^{2}}, where f is a homeomorphism between {\mathbb{A}(r,R)} and {\mathbb{A}(r_{\ast},R_{\ast})} , which belongs to the Sobolev class {\mathscr{W}^{1,2}} . The minimizer is a certain generalized radial mapping, i.e., a mapping of the form {f(\lvert x\rvert\eta)=\rho(\lvert x\rvert)T(\eta)} , where T is a conformal mapping of the unit sphere onto itself and {\rho(t)={R_{\ast}}\bigl{(}\frac{r_{\ast}}{R_{\ast}}\bigr{)}^{{\frac{R(r-t)}{(% R-r)t}}}} . It should be noticed that, in this case, no Nitsche phenomenon occurs.