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

TL;DR

This paper studies harmonic maps between concentric annuli in three-dimensional space, minimizing a weighted Dirichlet energy, resulting in a generalized radial solution without the Nitsche phenomenon.

Contribution

It introduces a minimization of a weighted Dirichlet integral for homeomorphisms between annuli in b3, deriving a generalized radial mapping as the minimizer, and shows no Nitsche phenomenon occurs.

Findings

The minimizer is a generalized radial mapping of the form f(|x|b7)=c1(|x|)T(b7)

The minimizer belongs to the Sobolev space a4^{1,2}

No Nitsche phenomenon occurs in this setting

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 $\mathscr{F}[f] = \int_{\mathbf{A}(r,R)} \frac{\Vert Df\Vert^2} {|f|^2}$, 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 $f(|x|\eta)=\rho(|x|)T(\eta)$, where $T$ is a conformal mapping of the unit sphere onto itself. It should be noticed that in this case no Nitsche phenomenon occur.