Harmonic maps between two concentric annuli in 𝐑3

Abstract

Abstract Given two annuli 𝔸 ⁢ ( r , R ) {\mathbb{A}(r,R)} and 𝔸 ⁢ ( r ∗ , R ∗ ) {\mathbb{A}(r_{\ast},R_{\ast})} , in 𝐑 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 ⁢ [ f ] = ∫ 𝔸 ⁢ ( r , R ) ∥ D ⁢ f ∥ 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 𝔸 ⁢ ( r , R ) {\mathbb{A}(r,R)} and 𝔸 ⁢ ( r ∗ , R ∗ ) {\mathbb{A}(r_{\ast},R_{\ast})} , which belongs to the Sobolev class 𝒲 1 , 2 {\mathscr{W}^{1,2}} . The minimizer is a certain generalized radial mapping, i.e., a mapping of the form f ⁢ ( | x | ⁢ η ) = ρ ⁢ ( | x | ) ⁢ T ⁢ ( η ) {f(\lvert x\rvert\eta)=\rho(\lvert x\rvert)T(\eta)} , where T is a conformal mapping of the unit sphere onto itself and ρ ⁢ ( t ) = R ∗ ⁢ ( r ∗ R ∗ ) 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.