# Reifenberg Flatness and Oscillation of the Unit Normal Vector

@article{Bortz2017ReifenbergFA, title={Reifenberg Flatness and Oscillation of the Unit Normal Vector}, author={Simon Bortz and Max Engelstein}, journal={arXiv: Classical Analysis and ODEs}, year={2017} }

We show (under mild topological assumptions) that small oscillation of the unit normal vector implies Reifenberg flatness. We then apply this observation to the study of chord-arc domains and to a quantitative version of a two-phase free boundary problem for harmonic measure previously studied by Kenig-Toro.

## 3 Citations

Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries

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- 2019

We show that given a domain $\Omega\subseteq \mathbb{R}^{d+1}$ with uniformly non-flat Ahlfors $s$-regular boundary and $s\geq d$, the dimension of its harmonic measure is strictly less than $s$.

Two phase free boundary problem for Poisson kernels

- MathematicsIndiana University Mathematics Journal
- 2022

We provide a potential theoretic characterization of vanishing chord-arc domains under minimal assumptions. In particular we show that, in the appropriate class of domains, the oscillation of the…

Harmonic Measure and the Analyst's Traveling Salesman Theorem

- Mathematics
- 2019

We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose…

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