Behavior near the boundary of positive solutions of second order parabolic equations. II

Fabes, E.; Safonov, M.; Yuan, Yu

doi:10.1090/S0002-9947-99-02487-3

Behavior near the boundary of positive solutions of second order parabolic equations. II
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by E. B. Fabes, M. V. Safonov and Yu Yuan PDF

Trans. Amer. Math. Soc. 351 (1999), 4947-4961 Request permission

Abstract:

A boundary backward Harnack inequality is proved for positive solutions of second order parabolic equations in non-divergence form in a bounded cylinder $Q=\Omega \times \left (0,T\right )$ which vanish on $\partial _xQ=\partial \Omega \times \left (0,T\right )$, where $\Omega$ is a bounded Lipschitz domain in $\mathbb {R}^n$. This inequality is applied to the proof of the Hölder continuity of the quotient of two positive solutions vanishing on a portion of $\partial _xQ.$

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