具有麦克斯韦边界条件的动力学福克 - 普朗克方程
Kinetic Fokker-Planck equations with Maxwell boundary conditions
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中文总结 AI 辅助
研究具有麦克斯韦边界条件的线性动力学福克 - 普朗克方程解的边界正则性理论，通过调节系数\(\alpha\in(0,1)\)解决中间区域问题，证明解的赫尔德连续性及建立特定阶边界正则性，还发展了可扩展的统一方法。
中文摘要 AI 辅助
我们为具有麦克斯韦边界条件的线性动力学福克 - 普朗克方程的解发展了边界正则性理论。这些条件通过调节系数\(\alpha\in[0,1]\)在漫反射和镜面反射之间进行插值。现有文献限于\(\alpha = 0\)和\(\alpha = 1\)的极端情况，我们解决了整个中间区域\(\alpha\in(0,1)\)。具体而言，我们表明当系数仅为一致椭圆时解是赫尔德连续的。此外，对于足够光滑的系数，我们建立了高达掠射集的\(\frac{3}{\pi}\arccos(\frac{\alpha}{2}) - 1\)阶的边界正则性，该指数是最优的。超越麦克斯韦条件，我们发展了一种统一方法，可扩展到包括超弹性碰撞在内的广泛反射边界条件类。
英文摘要
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 $α\in [0,1]$. While existing literature is restricted to the extreme cases $α= 0$ and $α= 1$, we resolve the entire intermediate regime $α\in (0,1)$. 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 $\frac{3}π \arccos(\fracα{2}) - 1$ 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.
