Sharp-Interface Limits of Periodic Phase-Transition Equations

Definitions and mathematical statement

Let $d$ be the spatial dimension, with no restriction on positive integer dimensions specified in the supplied record. Let $a:{\mathbb{R}}^d\to{\mathbb{R}}^{d\times d}$ be a uniformly elliptic matrix field that is ${\mathbb{Z}}^d$-periodic, meaning that \[ a(y+k)=a(y)\qquad (k\in{\mathbb{Z}}^d). \] The record does not specify the regularity, symmetry, or precise boundedness convention for $a$. Let $W:{\mathbb{R}}\to{\mathbb{R}}$ be the stated double-well potential, with global minima at $-1$ and $1$; in particular, \[ W(-1)=W(1)=\inf_{s\in{\mathbb{R}}}W(s). \] Here $W'$ denotes the derivative appearing in the equation; the source record does not give further regularity, growth, or nondegeneracy assumptions on $W$.

For each $\varepsilon>0$, let $u_\varepsilon:{\mathbb{R}}^d\times[0,\infty)\to{\mathbb{R}}$ solve the initial-value problem \[ \begin{cases} \partial_tu_\varepsilon-\operatorname{div}_x\!\left(a(x/\varepsilon)D_xu_\varepsilon\right)+\varepsilon^{-2}W'(u_\varepsilon)=0, & (x,t)\in{\mathbb{R}}^d\times(0,\infty),\\ u_\varepsilon(x,0)=u_0(x),&x\in{\mathbb{R}}^d. \end{cases} \] Here $D_x$ and $\operatorname{div}_x$ denote the spatial gradient and divergence. The datum $u_0$ is written without dependence on $\varepsilon$, as in the supplied statement. Its admissible class and the meaning of solution and initial trace are unspecified. The spatial domain is the whole space; no boundary condition or behavior at spatial infinity is supplied.

Problem. Determine the limiting behavior of $u_\varepsilon$ as $\varepsilon\to0^+$. The record does not specify a convergence topology, a class of admissible initial data, whether convergence is sought for the entire family or only subsequences, or a proposed limiting evolution. These omissions prevent a fully quantified convergence assertion. The title indicates a sharp-interface target, but no limiting interface or interface solution concept is defined in the supplied statement.

The original statement additionally asks for the limit for a periodic mean-curvature-type equation. That equation and its hypotheses are absent from the supplied record, so this additional part cannot be formulated precisely here.

Short English statement

Determine the small-$\varepsilon$ behavior of the stated phase-transition equation in a periodic uniformly elliptic medium, with fixed initial datum $u_0$. The source leaves the admissible data, solution concept, convergence mode, and limiting interface law unspecified. It also mentions a second, unspecified periodic mean-curvature-type equation.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; supplied permanent ID $30004655$, exact source problem code catalogue entry, title Sharp-Interface Limits of Periodic Phase-Transition Equations. The supplied collection is Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2021.
• Supplied report citation: Homogenization Theory: Periodic and Beyond (2022), Oberwolfach Reports, pp. 791--852, DOI 10.4171/owr/2021/14: https://doi.org/10.4171/owr/2021/14. The same reference is listed among the literature sources as Oberwolfach Report 14/2021. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The report contents have not been independently consulted for this draft.
• Supplied literature source: Morfe, Homogenization of Interface Motions, https://doi.org/10.6082/uchicago.4837.
• Supplied literature source: Morfe, Allen--Cahn with periodic mobility, https://arxiv.org/abs/2011.15012.
• Supplied literature assessment, checked on 2026-08-22: the record labels the problem partially_solved. It reports that homogenized interface laws and sharp-interface limits are proved for laminar media, restricted initial data, and periodic mobility/dissipation models, while the full periodic uniformly elliptic phase-transition problem as stated remains unresolved in all dimensions and data. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified, accompanied by a claim that the clean statement was checked against the official report and preserves its hypotheses, notation, and target. This is a registry verification claim, not verification performed for this draft; the missing mean-curvature-type equation remains unresolved.