Connected Concentration Sets and Solution Counts

Definitions and mathematical statement

The supplied record contains two revised questions about the singularly perturbed equation and energy functional of the report. Let $u_\varepsilon$ denote its solutions, let $E_\varepsilon[u_\varepsilon]$ be their energy, and let $N$ and $m$ be the dimensional parameters used there. The equation, functional, domain, boundary conditions, and admissible ranges of $N$ and $m$ are not included.

1. Suppose a family satisfies \[ \exists C>0\ \exists\varepsilon_0>0\ \forall\varepsilon\in(0,\varepsilon_0):\qquad E_\varepsilon[u_\varepsilon]\le C\varepsilon^{N-m}, \] where $C$ is independent of $\varepsilon$, and suppose its concentration set is connected. Must the limiting concentration set have dimension $m$? The source does not define concentration set, distinguish connectedness before and after passage to a limit, prescribe the mode of convergence, or specify the notion of dimension. The statement therefore retains these terms as report-dependent data; it does not replace dimension $m$ by an upper bound or by an assertion of smooth manifold structure.
2. What is the optimal asymptotic bound on the number of solutions as $\varepsilon\downarrow0$? If $\mathcal S_\varepsilon$ denotes the intended solution class and $\sim$ its intended notion of equivalence, the counted quantity would be $\#(\mathcal S_\varepsilon/\!\sim)$. Neither this class nor equivalence, finiteness, or the direction of the desired bound is specified. The record does not say whether the count is restricted to the energy-bounded, connected-concentration families in the first question or concerns a broader solution class.

The supplied verification says these are the two revised open questions following a negative answer to a preceding problem. That preceding problem and its counterexample are not supplied, so they cannot be reconstructed here. The displayed upper energy bound and connectedness are not presented as a self-contained universal assertion for arbitrary energy functionals: the report's missing PDE and concentration hypotheses are essential.

Short English statement

The first question asks whether a connected concentration set with the stated energy scaling must limit to an $m$-dimensional set. The second asks for the optimal growth of the number of solutions. Both require the report's missing equation, concentration definitions, and counting conventions.

Sources

• Local UnsolvedMath registry, permanent ID 30000257, code catalogue entry, "Connected Concentration Sets and Solution Counts." Proposed year 2005; proposer not specified.
• Topological and Variational Methods for Differential Equations (2006), Oberwolfach Reports, pp. 1601--1678, DOI 10.4171/owr/2005/29; original problem report https://doi.org/10.4171/owr/2005/29. The displayed year 2006 differs from the DOI and proposed year 2005.
• Supplied follow-up report with revised questions: https://ems.press/content/serial-article-files/46004. No further bibliographic details are supplied.
• Statement status corrected_verified: the supplied verification says the official report was checked directly and both revised questions following a negative answer to the preceding problem were confirmed. This is source metadata rather than an independent verification by this draft.
• Literature triage checked 2026-08-21: open. Its assessment says the revised questions are explicitly open in the follow-up report and the supplied search found no general resolution. Created 2026-08-25; updated 2026-09-16. Current status and cited claims require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report provenance remains as supplied. Unreviewed statement-only draft.