Radial symmetry of minimizers to the weighted p-Dirichlet energy

Abstract

Let $\mathbb {A}=\{z: r< |z|<R\}$ and $\mathbb {A}^*=\{z: r^*<|z|<R^*\}$ be annuli in the complex plane. Let $p\in [1,2]$ and assume that $\mathcal {H}^{1,p}(\mathbb {A},\mathbb {A}^*)$ is the class of Sobolev homeomorphisms between $\mathbb {A}$ and $\mathbb {A}^*$, $h:\mathbb {A}\xrightarrow []{{}_{\!\!\text {onto\,\,}\!\!}}\mathbb {A}^*$. Then we consider the following Dirichlet type energy of h:

$$\begin{aligned}\mathscr {F}_p[h]=\int _{\mathbb {A}}\frac{\Vert Dh\Vert ^p}{|h|^p}, 1\leqslant p\leqslant 2.\end{aligned}$$

We prove that this energy integral attains its minimum, and the minimum is a certain radial diffeomorphism $h:\mathbb {A}\xrightarrow []{{}_{\!\!\text {onto\,\,}\!\!}}\mathbb {A}^*$, provided a radial diffeomorphic minimizer exists. If $p>1$ then such diffeomorphism exists always. If $p=1$, then the conformal modulus of $\mathbb {A}^*$ must not be greater or equal to $\pi /2$. This curious phenomenon is opposite to the Nitsche type phenomenon known for the standard Dirichlet energy.