Radiall symmetry of minimizers to the weighted $p-$Dirichlet energy

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