The Divergence Theorem for Divergence Measure Vectorfields on Sets with Fractal Boundaries

摘要

A divergence measure vectortield is an R~n valued measure on an open subset U of R~n whose weak divergence in U is a (signed) measure. The paper uses the product rule for the product of the divergence measure by a function from W~(I,__)(U) established in Silhav_ [Silhav_, M., submitted, 2007] to prove the divergence theorem for the divergence measure vectorfields on bounded open sets U. It is shown that the surface integral of the normal component of the vectortield occurring in the classical divergence theorem has to be replaced by a continuous linear functional on the space of Lipschitz functions on the boundary; the volume integral contains the duality pairing occurring in the product rule. The boundary of U is arbitrary, it can be even fractal in the sense that the normal to au cannot be defined.