Minimization of Dirichlet energy of j−degree mappings between annuli

Let $A$ and $A_{\ast }$ be circular annuli in the complex plane, and consider the Dirichlet energy integral of $j$-degree mappings between $A$ and $A_{\ast }$. We aim to minimize this energy integral. The minimizer is a $j$-degree harmonic mapping between the annuli $A$ and $A_{\ast }$, provided it exists. If such a harmonic mapping does not exist, then the minimizer is still a $j$-degree mapping which is harmonic in $A^{\prime }\subset A$, and it is a squeezing mapping in its complementary annulus $A^{\prime \prime }=A\setminus A^{\prime }$. This result is an extension of a certain result by Astala et al. (2010).