Abstract
Abstract Given two annuli đ¸ ⢠( r , R ) {\mathbb{A}(r,R)} and đ¸ ⢠( r â , R â ) {\mathbb{A}(r_{\ast},R_{\ast})} , in đ 3 {\mathbf{R}^{3}} equipped with the Euclidean metric and the weighted metric | y | - 2 {\lvert y\rvert^{-2}} , respectively, we minimize the Dirichlet integral, i.e., the functional ⹠⢠[ f ] = ⍠đ¸ ⢠( r , R ) ⼠D ⢠f ⼠2 | f | 2 , \mathscr{F}[f]=\int_{\mathbb{A}(r,R)}\frac{\lVert Df\rVert^{2}}{\lvert f\rvert% ^{2}}, where f is a homeomorphism between đ¸ ⢠( r , R ) {\mathbb{A}(r,R)} and đ¸ ⢠( r â , R â ) {\mathbb{A}(r_{\ast},R_{\ast})} , which belongs to the Sobolev class đ˛ 1 , 2 {\mathscr{W}^{1,2}} . The minimizer is a certain generalized radial mapping, i.e., a mapping of the form f ⢠( | x | ⢠Ρ ) = Ď â˘ ( | x | ) ⢠T ⢠( Ρ ) {f(\lvert x\rvert\eta)=\rho(\lvert x\rvert)T(\eta)} , where T is a conformal mapping of the unit sphere onto itself and Ď â˘ ( t ) = R â ⢠( r â R â ) R ⢠( r - t ) ( R - r ) ⢠t {\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.
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