Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains

Abstract

In the first part of this paper we give intrinsic characterizations of the classes of Lipschitz and C 1 domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domain is Lipschitz if and only if it has a continuous transversal vector field. We also show that if the geometric measure theoretic unit normal of the domain is continuous, then the domain in question is of class C 1 . In the second part of the paper, we study the invariance of various classes of domains of locally finite perimeter under biLipschitz and C 1 diffeomorphisms of the Euclidean space. In particular, we prove that the class of bounded regular SKT domains (previously called chord-arc domains with vanishing constant, in the literature) is stable under C 1 diffeomorphisms. A number of other applications are also presented.

Characterizing regularity of domains via the Riesz transforms on their boundaries

“…[24] in this regard). This being said, the limiting case α = 0 of the equivalence (a)⇔(b) in Theorem 1.1 requires replacing the space of continuous functions by the (larger) Sarason space VMO, of functions of vanishing mean oscillations (on ∂Ω, viewed as a space of homogeneous type, in the sense of Coifman-Weiss, when equipped with the measure σ and the Euclidean distance).…”

“…The following characterization of the class of C 1+α domains from [24] is going to play an important role for us here. Theorem 2.2.…”

“…Fix x ∈ ∂ * Ω and pick an arbitrary δ > 0. From Lemma 2.5 we know that there exist O x ⊂ (0, 1) of density 1 at 0 (i.e., satisfying (2.32)) and some r δ > 0 with the property that 24) it follows that there exists ε δ ∈ (0, r δ ) with the property that…”

Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains

“…It has been shown by Hofmann et al [34] that is a Lipschitz graph domain if and only if it has finite perimeter in the sense of De Giorgi (see [11,21,28]) and (i) there are continuous (or, equivalently, smooth) vector fields that are transversal to the boundary and (ii) the necessary condition ∂ = ∂ is fulfilled. An equivalent characterization obtained in [34] is as follows.…”

“…An equivalent characterization obtained in [34] is as follows. A bounded nonempty domain of finite perimeter for which ∂ = ∂ is a Lipschitz graph domain if and only if…”