Fluxes Across Parts of Fractal Boundaries

Abstract

This paper determines the flux F (q,T) of a physical quantity across a part T of the boundary of a ‘rough body.’ The latter term means that the (measure theoretic) boundary ∂B of the body B is fractal in the sense that the outer normal n to B is not defined for almost every point of ∂B with respect to the n − 1 dimensional Hausdorff measure \(\mathcal {H}^{n-1}\). The quantity is represented by a bounded measurable flux vectorfield q on \(\mathbb{R}^n\) with bounded distributional divergence. Cauchy’s formula for regular surfaces T, $$F(\user2{q}, T) = \int_T \user2{q}\cdot \user2{n}d{\mathcal H}^{n-1},$$ (*) cannot be used because it requires the normal n to the surface. F(q, T) is defined using the divergence theorem provided T is a “trace,” i.e., provided T = B ∩ ∂M where M is a properly normalized set of finite perimeter. The definition reduces to (*) if \({\mathcal H}^{n-1} (T) < \infty\). The set \({\mathcal B}(B)\) of all traces is a boolean algebra and F(q, ·) is additive on it. Basic properties of the functional F are examined. (1) It is shown that if \({\mathcal H}^{n-1}(\partial B) =\infty\), then F (q, ·) does not extend to a measure unless q is in some sense trivial. (2) It is proved that a rough body B can be approximated by a sequence Bk of sets of finite perimeter such that (*) holds in some limiting sense. (3) Consequences are derived of the situation when a given \(T \in {\mathcal B}(B)\) insulates under q in the sense that the flux through each trace S⊂ T vanishes. (4) Conditions are given on ∂B for the locality of F (so that the value F(q, T) depends on the values of q on T).