On the Existence of $H^1$ solutions for Stationary Linearized Boltzmann Equations in a Small Convex Domain

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