Non-divergence parabolic equations of second order with critical drift in Lebesgue spaces

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• 作者：Gong Chen ^{gc@math.uchicago.edu}
• 关键词：Second-order parabolic equations ; Harnack inequality ; Measurable coefficients ; Spectral gap
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：262
• 期：3
• 页码：2414-2448
• 全文大小：428 K
• 卷排序：262

文摘

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+1Q⊂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 [19].