Harmonic maps between two concentric annuli in ${\mathbf{R}}^3$

Abstract

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

14 pages, this version contains some missing details from the previous version