The Divergence Theorem for Unbounded Vector Fields

In the context of Lebesgue integration, we derive the divergence theorem for unbounded vector fields that can have singularities at every point of a compact set whose Minkowski content of codimension greater than two is finite. The resulting integration by parts theorem is applied to removable sets of holomorphic and harmonic functions. In the context of Lebesgue integration, powerful divergence theorems (alternatively called Gauss-Green theorems) have been proved in [5, 6, 11] for bounded vector fields. Using generalized Riemann integrals, divergence theorems for certain kinds of unbounded vector fields are established in [12] and [10]. These vector fields are allowed to exhibit a controlled growth to infinity in neighborhoods of points lying on small sets — planes of codimension greater than two in [12], and compact sets of finite Hausdorff measure of codimension greater than two in [10]. However, a closer look at these results shows that, due to the growth conditions imposed, they apply only to vector fields with a finite number of singular points (see Remark 2.4 below for details). In contrast, we obtain a divergence theorem for vector fields having singularities at all points of certain compact sets of finite upper Minkowski content of codimension greater than two (see the Main Theorem below). This is achieved by imposing growth conditions about exceptional sets, rather than about individual points of exceptional sets. We employ only the Lebesgue integral, and integrate over bounded BV sets. Our result generalizes substantially the divergence theorem of [11]. As in [11], we show here that mere integrability of divergence implies the Gauss-Green formula; the separate integrability of the relevant partial derivatives is not required. In view of this, we can employ vector fields whose coordinates are not BV functions (see the paragraph following Example 2.6 below). Whether Minkowski contents can be replaced by Hausdorff measures is unclear. The techniques used in [7, Section 3] for characterizing removable sets of linear partial differential equations in terms of Hausdorff measures yield results different from ours (cf. Theorems 4.2 and 4.3 below), and they do not appear directly applicable in our approach. Received by the editors August 11, 2005. 2000 Mathematics Subject Classification. Primary 26B20; Secondary 26B05, 28A75.