  In a general $C^1$ domain, we study the perturbative Cauchy theory for the
Boltzmann equation with Maxwell boundary conditions with an accommodation
coefficient $\alpha$ in $(\sqrt{2/3},1]$, and discuss this threshold. We
consider polynomial or stretched exponential weights $m(v)$ and prove
existence, uniqueness and exponential trend to equilibrium around a global
Maxwellian in $L^\infty_{x,v}(m)$. Of important note is the fact that the
methods do not involve contradiction arguments.
