A Rough Divergence Theorem

2025/06/11 by Ruf, Thomas
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.09978

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 ⊂ \R^m has finite norm.

Citations

• On smooth interior approximation of Sets of Finite Perimeter
• A theory of traces and the divergence theorem
• Divergence-Measure Fields: Gauss-Green Formulas and Normal Traces
• A General Theorem of Gauß Using Pure Measures
• Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws
• Divergence-Measure Fields and Hyperbolic Conservation Laws

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