Journal
AMERICAN JOURNAL OF MATHEMATICS
Volume 143, Issue 1, Pages 307-331Publisher
JOHNS HOPKINS UNIV PRESS
DOI: 10.1353/ajm.2021.0001
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Funding
- NSF [DMS-1200701]
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In this paper, we extend the classical Krylov-Safonov Harnack inequality to consider functions with a two-scale behavior that may not satisfy an infinitesimal equation. The results have a wide range of applications in settings such as discrete difference equations, nonlocal equations, homogenization, and quasi-minimal surfaces of Almgren.
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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