The radial symmetry of minimizers to the weighted Dirichlet energy in

Abstract

Let $$ and $$ be annuli in $$. Let $$, and assume that $$ is the class of Sobolev $$ homeomorphisms of $$ onto $$. Then, we consider the following Dirichlet-type energy of $$:

For general $$, we minimize the Dirichlet-type integral $$ throughout the class of radial mappings between given annuli, and this minimum always exists for $$. For $$, the image annulus cannot be too thick, which is opposite to the Nitsche-type phenomenon known for the standard Dirichlet energy, where the image annulus cannot be too thin.