Minimization of Dirichlet energy of $j-$degree mappings between annuli

TL;DR

This paper studies the minimization of Dirichlet energy for $j$-degree mappings between annuli, identifying harmonic minimizers or squeezing mappings depending on the existence of harmonic solutions.

Contribution

It extends previous results by characterizing energy minimizers as harmonic or squeezing mappings in the context of $j$-degree annuli mappings.

Findings

Minimizers are harmonic $j$-degree mappings when they exist.

If harmonic mappings do not exist, minimizers are squeezing mappings in the annulus.

The results generalize earlier work by Astala, Iwaniec, and Martin.

Abstract

Let $\mathbb{A}$ and $\mathbb{B}$ be circular annuli in the complex plane and consider the Dirichlet energy integral of $j-$degree mappings between $\mathbb{A}$ and $\mathbb{B}$. Then we minimize this energy integral. The minimizer is a $j-$degree harmonic mapping between annuli $\mathbb{A}$ and $\mathbb{B}$ provided it exits. If such a harmonic mapping does not exist, then the minimizer is still a $j-$degree mapping which is harmonic in $\mathbb{A}'\subset \mathbb{A}$ and it is a squeezing mapping in its complementary annulus $\mathbb{A}''=\mathbb{A}\setminus \mathbb{A}$. Such a result is an extension of the certain result of Astala, Iwaniec and Martin \cite{astala2010}.