Radial symmetry of minimizers to the weighted p-Dirichlet energy

Let \(\mathbb {A}=\{z: r< |z|<R\}\) $\mathbb{A}=\{z:r<|z|<R\}$ and \(\mathbb {A}^*=\{z: r^*<|z|<R^*\}\) ${\mathbb{A}}^{\ast }=\{z:r^{\ast }<|z|<R^{\ast }\}$ be annuli in the complex plane. Let \(p\in [1,2]\) $p\in [1,2]$ and assume that \(\mathcal {H}^{1,p}(\mathbb {A},\mathbb {A}^*)\) ${\mathcal{H}}^{1,p}(\mathbb{A},{\mathbb{A}}^{\ast })$ is the class of Sobolev homeomorphisms between \(\mathbb {A}\) $\mathbb{A}$ and \(\mathbb {A}^*\) ${\mathbb{A}}^{\ast }$ , \(h:\mathbb {A}\xrightarrow []{{}_{\!\!\text {onto\,\,}\!\!}}\mathbb {A}^*\) $h:\mathbb{A}\stackrel{{}_{\text{onto,,}}}{\to }{\mathbb{A}}^{\ast }$ . Then we consider the following Dirichlet type energy of h: \(\begin{aligned}\mathscr {F}_p[h]=\int _{\mathbb {A}}\frac{\Vert Dh\Vert ^p}{|h|^p}, 1\leqslant p\leqslant 2.\end{aligned}\) $$\begin{array}{c}\hfill {\mathcal{F}}_p[h]={\int }_{\mathbb{A}}\frac{{\Vert Dh\Vert }^p}{{|h|}^p},1⩽p⩽2.\end{array}$$ We prove that this energy integral attains its minimum, and the minimum is a certain radial diffeomorphism \(h:\mathbb {A}\xrightarrow []{{}_{\!\!\text {onto\,\,}\!\!}}\mathbb {A}^*\) $h:\mathbb{A}\stackrel{{}_{\text{onto,,}}}{\to }{\mathbb{A}}^{\ast }$ , provided a radial diffeomorphic minimizer exists. If \(p>1\) $p>1$ then such diffeomorphism exists always. If \(p=1\) $p=1$ , then the conformal modulus of \(\mathbb {A}^*\) ${\mathbb{A}}^{\ast }$ must not be greater or equal to \(\pi /2\) $\pi /2$ . This curious phenomenon is opposite to the Nitsche type phenomenon known for the standard Dirichlet energy.