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

Kinetic Fokker-Planck and Landau Equations with Specular Reflection Boundary Condition