  We establish existence of finite energy weak solutions to the kinetic
Fokker-Planck equation and the linearized Landau equation near Maxwellian, in
the presence of specular reflection boundary condition for general domains.
Moreover, by using a method of reflection and the $S_p$ estimate previously
established by the first and the last author, we prove regularity in the
kinetic Sobolev spaces $S_p$ and anisotropic H\"older spaces for such weak
solutions. Such $S_p$ regularity leads to the uniqueness of weak solutions.
