  In this article, we investigate the incoming boundary value problem for the
stationary linearized Boltzmann equations in $ \Omega \subseteq
\mathbb{R}^{3}$. For a $C^2$ bounded domain with boundary of positive Gaussian
curvature, the existence theory is established in $H^{1}(\Omega \times
\mathbb{R}^{3})$ provided that the diameter of the domain $\Omega$ is small
enough.
