Abstract
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.
On Liouville-type theorems for the 2D stationary MHD equations
