Generalized Harmonic Measures: Synchronized Approximation, Representation and Quantitative Measure Recovery
Year: 2026
Type: preprint
Abstract: We study synchronized boundary and kernel approximation for harmonic measure systems satisfying nested mean-value identities. Under boundary concentration along a common subsequence, joint limits yield representation on a common refinement, and gluing gives uniqueness on the minimal boundary. Neighborhood nonvanishing gives full-sequence synchronized selection. For a compact set whose removal from a bounded $C^{1,1}$ domain in dimension at least three leaves a connected domain, zero Newtonian capacity is equivalent to concentration of one normalized singular kernel, constructed from harmonic measures, on a subsequence of one relatively compact exhaustion. For such zero-capacity sets, capacitary estimates give quantitative recovery of finite signed source measures on every prescribed relatively compact exhaustion. On rectifiable curve networks in three dimensions, the logarithmic error bound is sharp for this recovery formula in the class of finite measures. Boundary splitting and finite weighted metric graphs illustrate the gluing conclusions.
Source: arXiv (Cornell University)
Authors: Liu, Duchao, Wang, Yunjie
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G2620303751
OpenAlex ID: W7214647920