Coincidence of Supports and Maxima in Infinity-Eigenvalue Limits

Definitions and mathematical statement

For a specified domain $D$ and a function $v:D\to{\mathbb{R}}$, define \[ \operatorname*{argmax}_{D}v :=\{x\in D: v(x)\geq v(y)\text{ for every }y\in D\}. \] Here $D$ is a placeholder for the domain intended by the source, whose hypotheses are not supplied. The symbols $u_\infty$, $f_\infty$, and $d_\Omega$ refer to objects in the stated limiting infinity-eigenvalue and optimal-transport problem. Their definitions, the relationship between $D$ and $\Omega$, and the topology and meaning of $\operatorname{spt}(f_\infty)$ are not specified in the supplied record. In particular, no definition of $d_\Omega$ or type of $f_\infty$ is imposed here.

The supplied question asks whether the inclusions \[ \operatorname{spt}(f_\infty) \subseteq \operatorname*{argmax}_{D}u_\infty \subseteq \operatorname*{argmax}_{D}d_\Omega \] can both be replaced by equalities: \[ \operatorname{spt}(f_\infty) =\operatorname*{argmax}_{D}u_\infty =\operatorname*{argmax}_{D}d_\Omega. \] This is a schematic transcription of the proposed target, not a complete quantified conjecture: the admissible domains, limiting constructions, normalizations, and quantification over possible limiting objects are missing. The inclusions themselves are supplied assertions and are not independently verified here.

Short English statement

The question asks whether the support of the limiting object $f_\infty$, the maximum set of $u_\infty$, and the maximum set of $d_\Omega$ coincide. The supplied record omits the setting and definitions needed for a precise conjecture and contains conflicting descriptions of the completeness of the source statement.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002286$, source problem code catalogue entry, titled "Coincidence of Supports and Maxima in Infinity-Eigenvalue Limits." The record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is identified, and the supplied proposed year is 2013.
• Mini-Workshop: The p-Laplacian Operator and Applications (2013), Oberwolfach Reports, pp. 433--482, DOI 10.4171/owr/2013/08. Supplied source URL and literature-source link, labeled "Oberwolfach Report 2013": https://doi.org/10.4171/owr/2013/08.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open but says the extracted record contains only an announcement of an open problem and no complete mathematical assertion, leaving it unresolved under the queue rule for unclear statements. This is supplied dated curation, not an independently verified literature assessment.
• The record separately labels the statement corrected_verified and reports that the official contribution explicitly asks whether both displayed inclusions are equalities. This supplied verification claim conflicts with the completeness assessment above; neither claim has been independently checked here.