Gauss–Green Boundaries of Infinite Electrical Networks

Definitions and mathematical statement

Let $G$ be an infinite electrical network. Write $\mathcal H_E$ for its energy space, namely the space associated with functions of finite network Dirichlet energy, with the quotient and completion conventions to be specified. Write $\Delta_G$ for the network Laplacian and $\operatorname{Harm}\subseteq\mathcal H_E$ for the subspace represented by finite-energy harmonic functions, that is, functions satisfying $\Delta_G h=0$ at every vertex. The supplied record does not specify the conductance hypotheses, scalar field, normalization of the energy and Laplacian, or precise construction of $\mathcal H_E$.

Boundary-identification problem. For an arbitrary infinite electrical network in the intended class, identify an appropriate boundary $\partial G$, including its underlying space, measurable structure, and measure, such that integration over this boundary represents the boundary term in the discrete Gauss--Green formula for the energy space. Explain the relationship of this representation to $\operatorname{Harm}$.

More explicitly, if $B_G(u,v)$ denotes the boundary term in the intended discrete Gauss--Green identity, the requested representation has the schematic form \[ B_G(u,v)=\int_{\partial G} b_G(u,v;\xi)\,\mathrm d\mu_G(\xi) \] for every admissible pair $(u,v)$. Here the boundary integrand $b_G$, the measure $\mu_G$, and the admissible class of pairs are part of the unresolved formulation; the displayed expression does not assert their existence or prescribe their form. In particular, the supplied record does not give the precise Gauss--Green identity, its convergence requirements, a boundary trace or normal derivative, or the criterion that makes a boundary appropriate or intrinsic. These omissions prevent a fully specified quantified representation statement.

Short English statement

Identify a boundary space and measure for an infinite electrical network that express the boundary contribution in its discrete Gauss--Green formula as an integral, and clarify the connection with finite-energy harmonic functions. The supplied formulation leaves the network hypotheses, admissible functions, and boundary integrand unspecified. It is supplied as an open question and remains a statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001119$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The project identifies data/unsolvedmath/problems.json and the supplemental catalog/problems.json as its local registries; file membership has not been independently checked here.
• Supplied report citation: Interplay of Analysis and Probability in Physics (2009), Oberwolfach Reports, pp. 3065--3138, DOI 10.4171/owr/2008/55. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2008/55. No individual proposer is named; the record gives proposed year 2008.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, says that the report develops the representation without identifying the boundary, and reports that no general identification was found. Its listed checked source is the original OWR report linked above. This is supplied dated curation, not an independently verified literature assessment.
• The supplied statement status is corrected_verified; its verification note reports comparison with the official Oberwolfach report and normalization of notation, context, and question form. That verification claim is retained as source metadata and has not been independently checked here.