Open Access
may 2014 Approximation of a Reifenberg-flat set by a smooth surface
Guy David
Bull. Belg. Math. Soc. Simon Stevin 21(2): 319-338 (may 2014). DOI: 10.36045/bbms/1400592628

Abstract

On montre que si l'ensemble $E \subset \Bbb R^n$ est bien approché, à l'échelle $r_0$, par des plans de dimension $d$, il existe une surface lisse $\Sigma_0$ de dimension $d$, qui est proche de $E$ à l'échelle $r_0$. Quand $E$ est Reifenberg-plat, ceci permet d'appliquer un résultat de G. David et T. Toro [Memoirs of the AMS 215 (2012), 1012], et de montrer que $E$ est l'image de $\Sigma_0$ par un homéomorphisme bi-Höldérien de $\Bbb R^n$. Si de plus $d=n-1$ et $E$ est compact et connexe, alors $\Sigma_0$ est orientable, et $\Bbb R^n \sm E$ a exactement deux composantes connexes que la construction ci-dessus permet d'approximer par l'intérieur par des domaines lisses.

We show that if the set $E \subset \Bbb R^n$ is well approximated at the scale $r_0$ by planes of dimension $d$, we can find a smooth surface $\Sigma_0$ of dimension $d$ which is close to $E$ at the scale $r_0$. 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 $\Bbb R^n$ that sends $\Sigma_0$ to $E$. If in addition $d=n-1$ and $E$ is compact and connected, then $\Sigma_0$ is orientable, and $\Bbb R^n \sm E$ has exactly two connected components, which we can approximate from the inside by smooth domains.

Citation

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Guy David. "Approximation of a Reifenberg-flat set by a smooth surface." Bull. Belg. Math. Soc. Simon Stevin 21 (2) 319 - 338, may 2014. https://doi.org/10.36045/bbms/1400592628

Information

Published: may 2014
First available in Project Euclid: 20 May 2014

zbMATH: 1294.53007
MathSciNet: MR3211019
Digital Object Identifier: 10.36045/bbms/1400592628

Subjects:
Primary: 28A75 , 49Q20

Keywords: orientability , Reifenberg flat sets , Reifenberg topological disk theorem , smooth approximation

Rights: Copyright © 2014 The Belgian Mathematical Society

Vol.21 • No. 2 • may 2014
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