Fluxes of mechanical quantities across fractal boundaries

This seminar deals with fluxes of quantities across parts of the boundary of a body R with fractal boundary. A broad interpretation of the notion of a fractal boundary is employed here to mean any bounded subset R of Rn whose boundary dR is so complicated that the outer normal n to R is not defined for a.e. point of dR. Here a.e. means almost everywhere with respect to the n-1 dimensional Hausdorff measure (area) Hn-1. For a fractal body, Hn-1(dR) is infinite, since otherwise R is a set of finite perimeter with an Hn-1 a.e. defined normal. The paper is concerned with determining the net flux F(q,T) of a scalar quantity (such as the heat flux) across a subset T of dR, where the quantity is represented by a continuous field of the flux vector q on Rn with integrable distributional divergence. The paper examines basic properties of the functional F: (1) On the negative side, it is shown that if Hn-1(dR) is infinite, then F(q, . ) does not extend to a measure unless q is in some sense trivial. (2) On the positive side, it is proved that each R can be approximated by a sequence Rj of sets of finite perimeter such that the classical Cauchy formula holds in some limiting sense. (3) Consequences are derived of the situation when a given T insulates under q in the sense that the flux through each trace S of T vanishes. (4) Conditions are given on dR for the locality of F (so that the value F(q,T) depends on the values of q on T). (5) Classes of flux vector fields are described to which F(q,T) can be extended.