A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients. (English) Zbl 1423.35086
Summary: We consider weak solutions of second-order partial differential equations of Kolmogorov-Fokker-Planck-type with measurable coefficients in the form
\[
\partial_tu+\langle v,\nabla_xu\rangle=\operatorname{div}_v(A(v,x,t)\nabla_vu)+\langle b(v,x,t),\nabla_vu\rangle+f,\quad (v,x,t)\in\mathbb{R}^{2n+1},
\]
where \(A\) is a symmetric uniformly positive definite matrix with bounded measurable coefficients; \(f\) and the components of the vector \(b\) are bounded and measurable functions. We give a geometric statement of the Harnack inequality recently proved by F. Golse et al. [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 19, No. 1, 253–295 (2019; Zbl 1431.35016)]. As a corollary, we obtain a strong maximum principle.
MSC:
| 35H20 | Subelliptic equations |
| 35K70 | Ultraparabolic equations, pseudoparabolic equations, etc. |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35Q84 | Fokker-Planck equations |
Keywords:
hypoelliptic equations; Kolmogorov equations; kinetic theory; Fokker-Planck equations; Harnack inequalityCitations:
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