Radial symmetry of minimizers to the weighted Dirichlet energy

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 λ depends on the independent variable z. We prove that for an increasing radial weight λ(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 λ(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.Recall that a Sobolev homeomorphism f ∈ W 1,1 loc (A * , C) has finite distortion if there is a measurable function K(z) 1, finite a.e., such that (1.2) |Df (z)| 2 2 K(z) J f (z) , J f (z) = det Df (z).