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

Abstract

Let $\mathbb{A}=\{z: r< |z|<R\}$ and $\A^\ast=\{z: r^\ast<|z|<R^\ast\}$ be annuli in the complex plane. Let $p\in[1,2]$ and assume that $\mathcal{H}^{1,p}(\A,\A^*)$ is the class of Sobolev homeomorphisms between $\A$ and $\A^*$, $h:\A\onto \A^*$. Then we consider the following Dirichlet type energy of $h$: $$\mathcal{F}_p[h]=\int_{\A(1,r)}\frac{\|Dh\|^p}{|h|^p}, \ \ 1\le p\le 2.$$ We prove that this energy integral attains its minimum, and the minimum is a certain radial diffeomorphism $h:\A\onto \A^*$, provided a radial diffeomorphic minimizer exists. If $p>1$ then such diffeomorphism exist always. If $p=1$, then the conformal modulus of $\A^\ast$ 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.