  Ellipsoidal BGK model (ES-BGK) is a generalized version of the BGK model
where the local Maxwellian in the relaxation operator of the BGK model is
extended to an ellipsoidal Gaussian with a Prandtl parameter $\nu$, so that the
correct transport coefficients can be computed in the Navier-Stokes limit. In
this work, we consider the existence and uniqueness of stationary solutions for
the ES-BGK model in a slab imposed with the mixed boundary conditions. One of
the key difficulties arise in the uniform control of the temperature tensor
from below. In the non-critical case $(-1/2<\nu<1)$, we utilize the property
that the temperature tensor is equivalent to the temperature in this range. In
the critical case, $(\nu=-1/2)$, where such equivalence relation breaks down,
we observe that the size of bulk velocity in $x$ direction can be controlled by
the discrepancy of boundary flux, which enables one to bound the temperature
tensor from below.
