Approximation of a Reifenberg-flat set by a smooth surface

2012/11/14 by David, Guy
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1211.3222

Abstract

We show that if E ı\Rⁿ is a Reifenberg flat set E of dimension d at scale r₀, we can find a smooth surface Σ₀ of dimension d which is close to E at the scale r₀. When E is a Reifenberg flat set, this allows us to apply a result of G. David and T. Toro [Memoirs of the AMS 215 (2012), 1012], and get a bi-Hölder homeomorphism of \Rⁿ that sends Σ₀ to E. If in addition d=n-1 and E is compact and connected, then Σ₀ is orientable, and \Rⁿ \sm E has exactly two connected components, which we can approximate from the inside by smooth domains.

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