2026/07/22 by Kyeongbae Kim, Marvin Weidner
#math.AP
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 α∈ [0,1]. While existing literature is restricted to the extreme cases α= 0 and α= 1, we resolve the entire intermediate regime α∈ (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 \frac3π \arccos(\fracα2) - 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.