  We study the situation in which the distribution of temperature a body is due
to its interaction with radiation. We consider the boundary value problem for
the stationary radiative transfer equation under the assumption of the local
thermodynamic equilibrium. We study the diffusion equilibrium approximation in
the absence of scattering. We consider absorption coefficient independent of
the frequency $ \nu $ (the so-called Grey approximation) and the limit when the
photons' mean free path tends to zero, i.e. the absorption coefficient tends to
infinity. We show that the densitive of radiative energy $ u $, which is
proportional to the fourth power of the temperature due to the Stefan-Boltzmann
law, solves in the limit an elliptic equation where the boundary value can be
determined uniquely in terms of the original boundary condition. We derive
formally with the method of matched asymptotic expansions the boundary
condition for the limit problem and we prove rigorously the convergence to the
solution of the limit problem with a careful analysis of some non-local
integral operators. The method developed here allows to prove all the results
using only maximum principle arguments.
