On the extension property of Reifenberg-flat domains

Abstract

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 ∞ 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].

New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data

“…This class of domains include all C 1 -domains, Lipschitz domains with small Lipschitz constants, and domains with fractal boundaries. There are also many published papers working on Reifenberg flat domains, their properties and applications, that can be found in [34,35,36,14,43] and related references therein.…”

Nonlinear obstacle problems with double phase in the borderline case

“…For further properties and applications on this, we refer to [14,33,38,39,44,46,48,49,53] and references therein.…”

Global Calderón–Zygmund theory for nonlinear elliptic obstacle problems with asymptotically regular nonlinearities

“…We refer to [10,11,14,18] and the references therein for a further discussion on Reifenberg flat domains. On the above definitions, R can be any positive number from a scaling invariance property of the problem (2.8), while δ is invariant under such a scaling.…”