Equilibrium Supports on Solids of Revolution

Definitions and mathematical statement

Let $0<s<1$. The record associates to a compact generating set $A$ a set of revolution $\Gamma(A)$ and its $s$-equilibrium measure $\mu_{s,\Gamma(A)}$. Under the usual three-dimensional Riesz interpretation, if $A\subseteq[0,\infty)\times{\mathbb{R}}$ is a meridian set, its revolution about the vertical axis is \[\Gamma(A)=\{(r\cos\theta,r\sin\theta,z):(r,z)\in A,\ 0\leq\theta\leq2\pi\}.\] For a compact set $K$, a Riesz $s$-equilibrium measure is a Borel probability measure minimizing \[I_s(\nu)=\iint_{K\times K}|x-y|^{-s}\,d\nu(x)\,d\nu(y),\qquad \nu\in\mathcal P(K).\] The intended admissible sets must permit the equilibrium measure in question, in particular a finite-energy probability measure in this interpretation. These ambient-space, axis, and finite-energy conventions are not specified in the extract and are stated conditionally, not added as verified source hypotheses. Write $B_{\mathrm{out}}(\Gamma(A))$ for the source's outer boundary. Its precise definition is also absent. The requested assertion, retaining this source-specific notation, is \[\forall s\in(0,1)\ \exists\text{ an admissible compact }A:\quad \operatorname{supp}\mu_{s,\Gamma(A)}\subsetneq B_{\mathrm{out}}(\Gamma(A)).\] Here the support is the closed set of points every neighborhood of which has positive measure; proper inclusion requires both containment and at least one omitted boundary point. The set $A$ may depend on $s$; a single set working for all $s$ is not requested. The supplied 2026-08-21 triage labels the problem partially solved and reports a nontrivial proved range including examples for $0<s<1/3$, while saying the full existence assertion for every $0<s<1$ is not established by the literature located.

Short English statement

For each exponent $0<s<1$, ask for a compact set of revolution whose equilibrium measure omits part of its outer boundary while remaining supported on that boundary. The generating set may depend on the exponent. The extract leaves the admissible generating sets and the precise meaning of outer boundary unstated.

Sources

• Local UnsolvedMath registry, permanent ID 30000729, source problem catalogue entry, Equilibrium Supports on Solids of Revolution, proposed year 2007; no individual proposer supplied.
• Supplied citation: Mini-Workshop: Geometric Measure Theoretic Approaches to Potentials on Fractals and Manifolds (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/19, pp. 1027--1072, https://doi.org/10.4171/owr/2007/19. Supplied literature label: original Oberwolfach report.
• Supplied literature label: Riesz energy on surfaces of revolution, https://doi.org/10.1063/1.2817823.