Pinning Thresholds for Forced Interfaces

Definitions and mathematical statement

The supplied record uses $u(x,t)$ for an interface height, $t$ for time, and $F$ for a forcing strength. It does not specify the evolution equation, the spatial domain, the admissible forcing strengths, or the initial and boundary conditions.

For an interface evolution corresponding to forcing strength $F$, the source describes pinning at a spatial point $x$ by \[ \lim_{t\to\infty}u(x,t)<+\infty, \] intended to mean that the limit exists and is finite, and describes absence of pinning at $x$ by \[ \lim_{t\to\infty}u(x,t)=+\infty. \] The question is to determine which forcing strengths satisfy the first condition and which satisfy the second.

These are the source's stated criteria, rather than a fully specified mathematical problem. In particular, the record does not state whether the conditions must hold at a fixed point, at every spatial point, or in another spatial sense. Although the background concerns random media, neither a probability model nor a probabilistic quantifier is supplied. Existence of the displayed limits and exhaustiveness of the two alternatives are not asserted by any supplied hypotheses. A precise classification therefore requires the missing evolution model and quantifiers.

Short English statement

Determine the forcing strengths for which the interface height converges to a finite limit and those for which it tends to $+\infty$. The supplied record leaves the interface equation, spatial setting, initial and boundary data, and spatial and probabilistic quantifiers unspecified, so the classification problem remains incompletely formulated.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001342$, title Pinning Thresholds for Forced Interfaces, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2009$ and names no proposer. Its label corrected_verified is accompanied only by the explanation that mathematical delimiters were added without changing the content; it does not establish completeness of the formulation.
• Supplied report citation: PDE and Materials (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/42, pp. 2335--2404. https://doi.org/10.4171/owr/2009/42. The background describes these reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or named author is supplied.
• Supplied literature source, labelled "pinning of interfaces in random media": https://arxiv.org/abs/0911.4254.
• Supplied literature source, labelled "zero-mean random medium": https://doi.org/10.4171/IFB/455.
• Supplied literature source, labelled "infinite pinning": https://arxiv.org/abs/2007.07605.
• Supplied dated curation, checked 2026-08-21: the record labels the problem partially solved and reports that pinning and depinning have been proved in several dimensions and force regimes, including later zero-mean and infinite-pinning results, while the full phase diagram remains unsettled. This is the supplied literature triage derived from OWR curation, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.