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arXiv · 1803.10183

Quasi-Harnack inequality

Daniela De Silva·Ovidiu Savin
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Abstract

In this paper we obtain some extensions of the classical Krylov-Safonov Harnack inequality. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. We require that at scale larger than some $r_0>0$ (small) the functions satisfy the comparison principle with a standard family of quadratic polynomials, while at scale $r_0$ they satisfy a Weak Harnack type estimate. We also give several applications of the main result in very different settings such as discrete difference equations, nonlocal equations, homogenization and the quasi-minimal surfaces of Almgren.

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Daniela De Silva, Ovidiu Savin. 2018-03-27. Quasi-Harnack inequality. https://arxiv.org/abs/1803.10183

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