  The existence and stability of the Landau equation (1936) in a general
bounded domain with a physical boundary condition is a long-outstanding open
problem. This work proves the global stability of the Landau equation with the
Coulombic potential in a general smooth bounded domain with the specular
reflection boundary condition for initial perturbations of the Maxwellian
equilibrium states. The highlight of this work also comes from the
low-regularity assumptions made for the initial distribution. This work
generalizes the recent global stability result for the Landau equation in a
periodic box (KGH-2016). Our methods consist of the generalization of the
wellposedness theory for the Fokker-Planck equation (HJV-2014, HJJ-2018) and
the extension of the boundary value problem to a whole space problem, as well
as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic
Fokker-Planck equations (GIMV-2016) and the Morrey estimates
(Polidoro-Ragusa-1998) to further control the velocity derivatives, which
ensures the uniqueness. Our methods provide a new understanding of the grazing
collisions in the Landau theory for an initial-boundary value problem.
