Kinetic Fokker-Planck equations with Maxwell boundary conditions

We develop the boundary regularity theory for solutions to linear kinetic Fokker-Planck equations with Maxwell boundary conditions.

These conditions interpolate between diffuse and specular reflection via an accommodation coefficient $\alpha \in [0,1]$.

While existing literature is restricted to the extreme cases $\alpha = 0$ and $\alpha = 1$, we resolve the entire intermediate regime $\alpha \in (0,1)$.

Specifically, we show that solutions are Hölder continuous if the coefficients are merely uniformly elliptic.

Furthermore, for sufficiently smooth coefficients, we establish boundary regularity of order $\frac{3}{\pi} \arccos(\frac{\alpha}{2}) - 1$ up to the grazing set.

This exponent is optimal.

Beyond Maxwell conditions, we develop a unified approach that extends to a broad class of reflection boundary conditions, including super-elastic collisions.