The radial symmetry of minimizers to the p$p$ weighted Dirichlet energy in R3$\mathbb {R}^3$

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 : We prove that this energy integral attains its minimum for and , and the minimum is a certain radial diffeomorphism . 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.