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

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 ${\mathbb{A}}^{\mathrm{\prime }}\subset \mathbb{A}$ and it is a squeezing mapping in its complementary annulus ${\mathbb{A}}^{\mathrm{\prime }\mathrm{\prime }}=\mathbb{A}\setminus \mathbb{A}$. Such a result is an extension of the certain result of Astala, Iwaniec and Martin \cite{astala2010}.