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

In the first part of this article we give intrinsic characterizations of the classes of Lipschitz and C¹ 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 C¹. In the second part of the article, we study the invariance of various classes of domains of locally finite perimeter under bi-Lipschitz and C¹ 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 C¹ diffeomorphisms. A number of other applications are also presented. Acknowledgements and Notes. The work of the authors was supported in part by NSF grants DMS-0245401, DMS-0653180, DMS-FRG0456306, and DMS-0456861.