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arXiv · 2606.10185

Boundary Harnack estimates of optimal order for kinetic Fokker-Planck equations

Kyeongbae Kim·Marvin Weidner
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Abstract

We establish higher order boundary Harnack estimates for solutions to kinetic Fokker-Planck equations with absorbing incoming boundaries. Unlike classical elliptic and parabolic equations with Dirichlet data, we show that the quotient of two solutions for kinetic equations is not $C^{\infty}$ up to the boundary. Instead, we develop a general theory showing that, near the grazing set, the quotient of two solutions is $C^{3/2}$ if the domain and data are sufficiently smooth, and $C^{1,1}$ in the absence of source terms. These exponents are optimal.

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Kyeongbae Kim, Marvin Weidner. 2026-06-08. Boundary Harnack estimates of optimal order for kinetic Fokker-Planck equations. https://arxiv.org/abs/2606.10185

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