  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}.
