Regularity of Stationary Boltzmann equation in Convex Domains

Higher regularity estimate has been a challenging question for the Boltzmann equation in bounded domains. Indeed, it is well-known to have "the non-existence of a second order derivative at the boundary" in [15] even for symmetric convex domains such as a disk or sphere. In this paper we answer this question in the affirmative by constructing the $C^{1,\beta}$ solutions away from the grazing boundary, for any $\beta<1$, to the stationary Boltzmann equation with the non-isothermal diffuse boundary condition in strictly convex domains, as long as a smooth wall temperature has small fluctuation pointwisely.