Uniqueness of Condensation-Model Parameters

Definitions and mathematical statement

Fix $\beta>1$. Let $m_\beta>0$ satisfy \[ m_\beta=\tanh(\beta m_\beta), \qquad m^*=\sqrt{1-\beta^{-1}}. \] For $u_+\in(m^*,m_\beta)$, the supplied record considers a profile $q:(-\infty,0]\to{\mathbb{R}}$ and scalar parameters $h_*$ and $m_-$ satisfying \[ q(x)=\tanh\!\left(\beta(J*q)(x)+\beta h_*\right) \quad\text{for }x\leq0, \qquad \lim_{x\to-\infty}q(x)=m_-, \qquad q(0)=u_+. \] The notation $J*q$ is retained from the source record. The record does not specify the kernel $J$, its assumptions, or how this convolution is defined for a profile given only on $( -\infty,0]$. It also does not specify profile regularity, monotonicity, or any additional restrictions on the parameters. Consequently, these displayed conditions do not yet define a complete boundary-value problem.

Conjectural uniqueness assertion, conditional on the missing specifications. For every fixed $\beta>1$ and every $u_+\in(m^*,m_\beta)$, any two admissible triples $(q_1,h_{*,1},m_{-,1})$ and $(q_2,h_{*,2},m_{-,2})$ satisfying the displayed conditions, with the same kernel and convolution prescription, satisfy \[ h_{*,1}=h_{*,2} \quad\text{and}\quad m_{-,1}=m_{-,2}. \] This asserts uniqueness of the parameter pair; it does not assert uniqueness of the profile.

The supplied mathematical statement asks for this uniqueness. The supplied background additionally describes an all-parameter existence and uniqueness conjecture: for every such $u_+$, there is exactly one admissible parameter pair for which a profile exists. Existence is an additional assertion, and the record does not resolve whether it is part of the intended target. Neither formulation can be made unconditional until the missing kernel, boundary prescription, and admissibility hypotheses are recovered.

Short English statement

For each prescribed boundary value $u_+\in(m^*,m_\beta)$, ask whether the half-line profile equation determines a unique field $h_*$ and limiting value $m_-$. The kernel and half-line convolution are unspecified. The background also includes existence for every boundary value, whereas the displayed question asks only about uniqueness. This is a statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002928$, title Uniqueness of Condensation-Model Parameters, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2015 and no named proposer.
• Supplied source 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. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "Oberwolfach Report on the stationary problem": https://doi.org/10.4171/owr/2015/35.
• Supplied literature source, labeled "De Masi--Presutti--Tsgkarogiannis, stationary Stefan problem": https://doi.org/10.1007/s00205-011-0423-1.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open, reports existence for some parameter values, and describes the full all-parameter existence and uniqueness assertion as conjectural, with no authoritative proof located by that curation. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked to restore the half-line equation, boundary conditions, parameter range, and definitions. This is supplied verification metadata; no source consultation or mathematical verification is claimed here.