Jenny Harrison

The Gauss-Green theorem for fractal boundaries

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

Continuity of the integral as a function of the domain

We present here the fundamentals of a theory of domains that offers unifying techniques and terminology for a number of different fields. Using direct, geometric methods, we develop integration over p-dimensional domains in n-dimensional Euclidean space ⩄n, replacing the method of parametrization of a domain with the method of approximation in Banach spaces. We prove basic results needed for a theory of integration — continuity of the integral as a function of its domain and integrand (Corollary 4.8) and a generalization of Stokes’s theorem (Corollary 4.14).

Isomorphisms of differential forms and cochains

This paper proves an isomorphism theorem for cochains and differential forms, before passing to cohomology. De Rham’s theorem is a consequence. This leads to an extension of much of calculus and homology theory to nonsmooth domains, called chainlets,and makes available combinatorial techniques for smooth domains that limit to the classic analytic methods. We find maximal subspaces of L1forms that satisfy Stokes’s theorem for domains of chainlets giving a measurable, as well as optimal, extension of the theory.

Existence and soap film regularity of solutions to Plateau’s problem

Abstract Plateau’s problem is to find a surface with minimal area spanning a given boundary. Our paper presents a theorem for codimension one surfaces in ℝ n

On Plateau's Problem for Soap Films with a Bound on Energy

{\mathbb{R}^{n}}

Cartan’s magic formula and soap film structures

in which the usual homological definition of span is replaced with a novel algebraic-topological notion. In particular, our new definition offers a significant improvement over existing homological definitions in the case that the boundary has multiple connected components. Let M be a connected, oriented compact manifold of dimension n - 2

Operator Calculus of Differential Chains and Differential Forms

{n-2}

General methods of elliptic minimization

and 𝔖

Solutions to the Reifenberg Plateau problem with cohomological spanning conditions

{\mathfrak{S}}

Geometric Representations of Currents and Distributions

the collection of compact sets spanning M. Using Hausdorff spherical measure as a notion of “size,” we prove: There exists an X 0