Expository Metastability Heading for a Nonlocal Dirichlet Problem

Definitions and mathematical statement

Formulation status. The supplied record does not determine a precise mathematical problem. It retains the question: Can metastable states be stable in the stationary nonlocal Dirichlet problem described in the report?

The nonlocal equation, spatial domain, interaction kernel, parameters, unknown function, and Dirichlet data are not supplied. Neither the class of metastable states nor the meaning of stability is defined. In particular, a question about dynamical stability requires a specified evolution, a class of admissible perturbations, a topology, and a time regime; these are absent. Consequently, no quantified existence or stability assertion can be formulated from this record without adding hypotheses.

According to the supplied statement-verification note, this question is an expository title and motivation for an account proving existence of stationary metastable profiles, rather than a separate source-stated open problem. The specific unresolved uniqueness conjecture is represented by the distinct record OWR-13856-004; its statement is not supplied here and is not substituted for this record.

The supplied literature assessment reports construction of stationary metastable profiles in a restricted Kac--Ising setting and describes the broader dynamical-stability question as unresolved. These are supplied curation claims, not independently verified mathematical or literature conclusions. The registry labels this entry open while marking its statement unrecoverable; neither label supplies the missing formulation.

Short English statement

This entry preserves an expository question about stability of metastable states in a nonlocal Dirichlet problem. The supplied verification says it is not a separate source-stated open problem. The equation and stability definitions are missing, so a precise mathematical statement cannot be recovered from the supplied record.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent catalogue entry, source code catalogue entry, title Expository Metastability Heading for a Nonlocal Dirichlet Problem. The supplied record retains the original question for provenance, provides no clean statement, and marks the statement unrecoverable. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Interplay of Analysis and Probability in Applied Mathematics (2016), Oberwolfach Reports, pp. 1989--2064, DOI 10.4171/owr/2015/35. https://doi.org/10.4171/owr/2015/35. This is also listed among the literature sources as "Oberwolfach Report on metastable states." The report is cited by the record but its relevant text is not supplied here.
• Supplied literature source: De Masi--Presutti--Tsgkarogiannis, Fourier law and stationary Stefan problem. https://arxiv.org/abs/1009.4652.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the topic open, reports stationary metastable profiles in a restricted Kac--Ising setting, and describes the broader dynamical-stability question as unresolved. This is supplied dated curation derived from the OWR material, not independently verified here; its time-sensitive status and cited claims require independent verification.