  A basic question about regularity of Boltzmann solutions in the presence of
physical boundary conditions has been open due to characteristic nature of the
boundary as well as the non-local mixing of the collision operator. Consider
the Boltzmann equation in a strictly convex domain with the specular,
bounce-back and diffuse boundary condition. With the aid of a distance function
toward the grazing set, we construct weighted classical $C^{1}$ solutions away
from the grazing set for all boundary conditions. For the diffuse boundary
condition, we construct $W^{1,p}$ solutions for $1< p<2$ and weighted $W^{1,p}$
solutions for $2\leq p\leq \infty $ as well. On the other hand, we show second
derivatives do not exist up to the boundary in general by constructing
counterexamples for all boundary conditions.
