Cone Conditions and Properties of Sobolev Spaces

Many of the most common and useful properties of Sobolev spaces defined over a domain (open set) in Euclidean space require that the domain has a minimal degree of regularity. To this end, the domain is often assumed to satisfy a “cone condition.” For example, various imbeddings of Sobolev spaces into Lebesgue spaces or spaces of bounded continuous functions (the Sobolev imbedding theorem), and various interpolation inequalities such as those estimating D-norms of intermediate-order derivatives of functions in terms of such norms of higherand lower-order derivatives (the Ehrling-NirenbergGagliardo theorem), are commonly proved under the assumption that the domain satisfies a cone condition. Several versions of the cone condition have been used, but the most common (and weakest) is as follows: The domain 52 C [w” is said to satisfy the cone condition if each point x E Q is the vertex of a finite, right-spherical cone C, contained in D and congruent to a fixed such cone C. (C, is the union of all points on line segments from x to points of a ball not containing x.) Many proofs based on the cone condition depend heavily on geometric consequences of the condition-for example the consequence that !2 is a finite union of subdomains each of which is a union of parallel translates of a parallelepiped. It has been noticed, however, (see Edmunds and Evans [4]), that in certain arguments of a potential theoretic nature an obviously weaker measure theoretic version of the cone condition suffices.