A Rough Divergence Theorem
Thomas Ruf
TL;DR
The paper extends the divergence theorem to domains with inner boundaries by introducing the bounded fluctuation framework ${\mathcal{BF}^\infty}(U)$, which yields a dual surface functional ${\mu_f}$ that decomposes into interior, boundary, and infinity components. It establishes that the theorem holds if and only if the domain has finite anexometer, equivalently characterized by approximability of the domain indicator and related functions by smooth objects with uniformly bounded gradients; this is analyzed via a robust BF^ Infinity seminorm and a lower-semicontinuous auxiliary seminorm. The approach generalizes classical trace theories and connects to Anzellotti-type pairings while providing a flexible framework to handle rough data and highly irregular boundaries. The results open avenues for extending divergence-type results to fractal-like domains and inform potential numerical schemes through density and approximation results anchored in coarea, Sard, and CLT-type arguments.
Abstract
A generalized divergence theorem is established allowing for domains with inner boundaries. The normal trace of a rough integrand is not a Radon measure; rather, the boundary integral is expressed via a surface functional continuous with respect to the uniform convergence of integrands. We provide necessary and sufficient analytic and geometric conditions on the domain for the validity of the theorem. Central to this characterization is the introduction of the space of functions having bounded fluctuation, whose norm is precisely defined so that the divergence theorem holds if and only if the characteristic function $χ_U$ of the integration domain $U \subset \R^m$ has finite norm.