HÖLDER CONDITIONS IN THE DIVERGENCE THEOREM

Abstract

In the framework of Lebesgue integration and bounded sets of finite perime- ter, we present a straightforward proof of the divergence theorem for bounded vector fields satisfying Holder conditions on sets of appropriate Hausdorff measures. Given a compact BV set A ⊂ R n , the divergence theorem holds for every vector field v : A → R n that is continuous outside an H n−1 negligible set, and pointwise Lipschitz outside an H n−1 σ-finite set (9, Theorem 2.9). Since continuous and pointwise Lipschitz are extreme points of the scale represented by Holder conditions, it is natural to ask whether the divergence theorem remains valid under the following assumptions: for 0 <s< 1, the vector field v is pointwise Lipschitz outside an H n−1+s σ-finite (negligible) set E ⊂ A, and the s-Holder constant of v is zero (finite) at each x ∈ E. We use results of W.B. Jurkat (see Remark 2.5 below) to prove the divergence theorem which takes into account Holder conditions for all 0 ≤ s ≤ 1 simultaneously. One of the consequences, obtained by restricting the number s to 0 and 1, is a new and simpler proof of the divergence theorem cited above. Our starting point is the well-known divergence theorem for bounded BV sets and con- tinuously differentiable vector fields, which is assumed without proof. Modulo a few techni- calities resulting from the use of BV sets, the exposition is elementary. The crux of the proof involves only dyadic cubes. As an application, we present two theorems about integration by parts. Both are easy corollaries of the main result.