The Gauss-Green theorem for fractal boundaries

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v is the 1-vectorfield “dual” to ω: if ω = ∑ (−1)fi dx1 ∧ · · · ∧ ∧ dxi ∧ · · · ∧ dxn, then v = (f1, . . . , fn).) There has been considerable effort in the literature (e.g. [JK], [M], [P]) to extend this formula to permit integrands of less regularity by generalizing the Lebesgue integral. On the other hand, invariably the situations in which (1) holds require fairly strong hypotheses on the boundary ∂Ω, e.g. that it should have sigmafinite (n−1)-measure, or that the gradient of the characteristic function of Ω be a vector valued measure with finite total variation [F], [P]. However there is a natural way to expand the validity of (1) to much more general boundaries while still using the ordinary Lebesgue integral; this is the topic of the present paper. For the case of Lipschitz forms, the results of this paper follow readily from Whitney’s theory of flat chains [W2]. However his approach to the Gauss-Green theorem is not widely appreciated because he focused on chains and cochains, where effectively (1) is used to define the exterior derivative. In [HN] we extend Whitney’s method to treat the more general Holder case. (Only the case n = 2 is discussed