Continuous Minimizers for a Regional Fractional-Laplacian Functional

Definitions and mathematical statement

The supplied record identifies a Rayleigh--Faber--Krahn variational problem associated with the regional fractional Laplacian, but does not supply its functional, admissible class, constraints, dimension assumptions, or fractional-order range. Thus a fully specified variational problem cannot be reconstructed from this record alone.

The following is a schematic formulation of the intended question. For fixed problem data, let $\mathcal A$ denote the intended admissible class and let $\mathcal J$ denote the intended objective functional. A minimizer is an element $u_*\in\mathcal A$ satisfying \[ \mathcal J(u_*)=\inf_{v\in\mathcal A}\mathcal J(v). \] Write $C({\mathbb{R}}^n)$ for the real-valued functions continuous at every point of ${\mathbb{R}}^n$. If admissible elements are equivalence classes of functions, continuity must instead mean the existence of a representative in $C({\mathbb{R}}^n)$, with the equivalence relation determined by the missing variational setup.

The intended question is whether there exists at least one minimizer $u_*$ that is continuous on all of ${\mathbb{R}}^n$, in the preceding sense. This requires global continuity, including at any boundary of its support or underlying admissible domain. If the variational problem initially defines functions only on a domain, the required extension to ${\mathbb{R}}^n$ must be specified; the supplied record does not state an extension convention. The question concerns the existence of some continuous minimizer, not the continuity of every minimizer.

The symbols $\mathcal A$ and $\mathcal J$ are placeholders for missing source data, not definitions of a particular regional fractional-Laplacian problem. The supplied dated assessment reports attainment of the infimum and labels the continuity question open; these claims are not independently verified here.

Short English statement

Does the intended Rayleigh--Faber--Krahn problem for the regional fractional Laplacian have at least one minimizer continuous everywhere on ${\mathbb{R}}^n$? The supplied record omits the variational setup and parameter assumptions, so the formulation remains schematic and requires mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30005069$, source problem code catalogue entry; title: "Continuous Minimizers for a Regional Fractional-Laplacian Functional." The supplied record attributes the problem to the Oberwolfach Reports Open Problems collection and gives proposed year 2022; no individual proposer is supplied.
• Supplied source citation: "MFO-RIMS Tandem Workshop: Nonlocality in Analysis, Probability and Statistics (2023), Oberwolfach Reports," DOI 10.4171/owr/2022/15, pp. 709--715. Supplied source URL and literature-source link, labeled "Oberwolfach Report 15/2022": https://doi.org/10.4171/owr/2022/15. The original excerpt also contains the page marker 747. These year and page discrepancies remain unresolved.
• Supplied literature source: "Jin--Kriventsov--Xiong, regional fractional Laplacian," https://arxiv.org/abs/2108.07967. No full bibliographic citation or specific supporting result is supplied.
• Literature assessment dated exactly 2026-08-22: the record reports that the variational problem has an attained minimizer and labels the existence of a globally continuous minimizer open. This is supplied dated curation, not an independently verified assessment. The record describes this assessment as literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims. Its corrected_verified label and claim of checking the reconstruction against the official report are likewise supplied provenance claims, not verification performed here.