$\mathbf{R}^3$中同心环间的调和映射
Harmonic maps between two concentric annuli in $\mathbf{R}^3$
摘要：
给定两个环$\mathbf{A}(r,R)$和$\mathbf{A}(r_\ast, R_\ast)$，在分别具有欧几里德度量和加权度量$|y|^{-2}$的$\mathbf{R}^3$中，最小化Dirichlet积分，即泛函$\mathscr{F}[f] = \int_{\mathbf{A}(r,R)} \frac{\Vert Df\Vert^2} {|f|^2}$，其中$f$是属于Sobolev类$\mathscr{W}^{1,2}$的$\mathbf{A}(r,R)$和$\mathbf{A}(r_\ast,R_\ast)$之间的同胚.极小化器是某种广义径向映射，即$f(|x|η)=ρ(|x|)T(η)$形式的映射，其中$T$是单位球面到其自身上的保角映射。应该注意的是，在这种情况下，没有Nitsche现象发生。
英文摘要：
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 $\mathscr{F}[f] = \int_{\mathbf{A}(r,R)} \frac{\Vert Df\Vert^2} {|f|^2}$, where $f$ is a homeomorphism between $\mathbf{A}(r,R)$ and $\mathbf{A}(r_\ast,R_\ast)$, which belongs to the Sobolev class $\mathscr{W}^{1,2}$. The minimizer is a certain generalized radial mapping, i.e. a mapping of the form $f(|x|η)=ρ(|x|)T(η)$, where $T$ is a conformal mapping of the unit sphere onto itself. It should be noticed that in this case no Nitsche phenomenon occur.
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