Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 51-71

ON THE EXTENSION PROPERTY OF REIFENBERG-FLAT DOMAINS

Antoine Lemenant, Emmanouil Milakis and Laura V. Spinolo

Université Paris Diderot - Paris 7 - LJLL - CNRS, U.F.R. de Mathématiques
Bâtiment Sophie Germain, 75205 Paris Cedex 13, France; lemenant 'at' ljll.univ-paris-diderot.fr

University of Cyprus, Department of Mathematics & Statistics
P.O. Box 20537, Nicosia, CY-1678 Cyprus; emilakis 'at' ucy.ac.cy

Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche
via Ferrata 1, I-27100, Pavia, Italy; spinolo 'at' imati.cnr.it

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^\infty 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].

2010 Mathematics Subject Classification: Primary 49Q20, 49Q05, 46E35.

Key words: Reifenberg-flat sets, extension operators.

Reference to this article: A. Lemenant, E. Milakis and L.V. Spinolo: On the extension property of Reifenberg-flat domains. Ann. Acad. Sci. Fenn. Math. 39 (2014), 51-71.

Full document as PDF file

doi:10.5186/aasfm.2014.3907