# Radially symmetry of minimizers to the weighted Dirichlet energy

## TL;DR

This paper proves that for certain radial weights, the minimizers of the weighted Dirichlet energy between planar annuli are radially symmetric, simplifying the problem and extending known results to more general weights.

## Contribution

It establishes radial symmetry of energy minimizers for a broad class of radial weights, including increasing weights and those with conformally thin targets.

## Key findings

- Radial symmetry holds for increasing radial weights.
- Symmetry also holds when the target is conformally thin.
- Results extend previous symmetry results to more general weights.

## Abstract

We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight $\lambda$ depends on the independent variable $z$. We prove that for an increasing radial weight $\lambda(z)$ the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight $\lambda(z)$ we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight.