We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift −ut+Lu=−ut+∑ijaijDiju+∑biDiu=0(≥0,≤0) in some domain Q⊂Rn+1. We prove growth theorems and the interior Harnack inequality as the main results. In this paper, we will only assume the drift b is in certain Lebesgue spaces which are critical under the parabolic scaling but not necessarily to be bounded. In the last section, some applications of the interior Harnack inequality are presented. In particular, we show there is a “universal” spectral gap for the associated elliptic operator. The counterpart for uniformly elliptic equations of second order in non-divergence form is shown in .
|Number of pages||35|
|Journal||Journal of Differential Equations|
|State||Published - Feb 5 2017|
Bibliographical notePublisher Copyright:
© 2016 Elsevier Inc.
- Harnack inequality
- Measurable coefficients
- Second-order parabolic equations
- Spectral gap
ASJC Scopus subject areas
- Applied Mathematics