Weighted Condenser-Energy Minimizers

Definitions and mathematical statement

Let $I$ be a countable index set and let $\mathbf A=(A_i)_{i\in I}$ be the condenser in the supplied record, where every $A_i\subseteq{\mathbb{R}}^n$ is nonempty and closed, and $A_i\cap A_j=\varnothing$ whenever $i\ne j$. Let $\mathbf a=(a_i)_{i\in I}$ satisfy \[ a_i>0\quad(i\in I),\qquad \sum_{i\in I}a_i<\infty. \] Let $g\in C({\mathbb{R}}^n)$, and suppose there exist constants $c_1,c_2$ such that \[ 0<c_1<g(x)<c_2<\infty\qquad(x\in{\mathbb{R}}^n). \] Let $\mathbf f=(f_i)_{i\in I}$ be an admissible external field in the sense intended by the source.

The record denotes the constrained admissible class by $\mathcal E_{\mathbf f}^+(\mathbf A,\mathbf a,g)$ and its energy functional by $G_{\mathbf f}$. Following its description of the target as an infimum, write \[ G_{\mathbf f}(\mathbf A,\mathbf a,g) :=\inf_{\boldsymbol\mu\in\mathcal E_{\mathbf f}^+(\mathbf A,\mathbf a,g)} G_{\mathbf f}(\boldsymbol\mu). \] The supplied question is whether, for every choice of data satisfying the stated hypotheses, \[ G_{\mathbf f}(\mathbf A,\mathbf a,g)<\infty \quad\Longrightarrow\quad \exists\boldsymbol\lambda\in\mathcal E_{\mathbf f}^+(\mathbf A,\mathbf a,g) \ \text{such that}\ G_{\mathbf f}(\boldsymbol\lambda) =G_{\mathbf f}(\mathbf A,\mathbf a,g). \] This is a conditional restatement: the supplied record does not specify the interaction kernel, the energy formula, admissibility of the external field, or the precise definition of the constrained class and its mass constraints. These missing definitions must be recovered before this becomes a fully specified mathematical problem. No additional condenser hypotheses or lower bound on the infimum are imposed here.

Short English statement

For the supplied countable condenser, positive summable masses, bounded positive continuous weight, and admissible external field, does the constrained energy infimum have a minimizer whenever it is less than $+\infty$? The energy and admissibility definitions are missing from the record, so the formulation remains conditional and requires human mathematical review.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002165$, title "Weighted Condenser-Energy Minimizers," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is named; the supplied proposed year is 2012.
• Supplied source citation: Optimal and Near Optimal Configurations on Lattices and Manifolds (2013), Oberwolfach Reports, pp. 2429--2492, DOI 10.4171/owr/2012/40. Supplied literature label: "OWR report 40." https://doi.org/10.4171/owr/2012/40.
• Supplied literature reference labelled "Zorii": https://doi.org/10.1007/s11118-012-9279-8. No title or fuller author attribution is supplied.
• Supplied literature reference labelled "Zorii solvability criteria": https://arxiv.org/abs/1207.0604. No fuller bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. The supplied status is "partially solved." The assessment states that existence is established under substantial hypotheses, including perfect-kernel settings with solvability criteria, and that the broad infinite-dimensional question is not universally affirmative. The background identifies this as dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified. Its accompanying verification note attributes the condenser, field, mass, weight, and finite-energy hypotheses and the attainment question to the official report. This is a supplied verification claim; no independent source inspection is asserted here.