On the extension property of Reifenberg-flat domains

Abstract

We provide a detailed proof of the fact that any open set whose boundary is sufficiently flat in the sense of Reifenberg is also Jones-flat,and hence it admits an extension operator. We discuss various applications of this property, in particular we obtain L 1 estimates for the eigenfunctions of the Laplace operator with Neumann boundary conditions. We also compare different ways of measuring the distance between two Reifenberg-flat domains. These results are pivotal to the quantitative stability analysis of the spectrum of the Neumann Laplacian performed in (27).