Inserted: 12 sep 2012
Last Updated: 10 feb 2015
Journal: Annales Academiae Scientiarum Fennicae Mathematica
We provide a detailed proof of the fact that any domain which is sufficiently flat in the sense of Reifenberg is also Jones-flat, and hence it is an extension domain. We discuss various applications of this property, in particular we obtain $L^\infty$ estimates for the eigenfunctions of the Laplace operator with Neumann boundary conditions. We also compare different ways of measuring the "distance" between two sufficiently close Reifenberg-flat domains. These results are pivotal to the quantitative stability analysis of the spectrum of the Neumann Laplacian performed in a second paper by the same authors intituled "Spectral Stability Estimates for the Dirichlet and Neumann Laplacian in rough domains".
Keywords: Reifenberg-flat domains, Extension domains