Gradient-Constrained Ginzburg-Landau Minimizers

Definitions and mathematical statement

Let $N\in{\mathbb{Z}}$ satisfy $2\leq N\leq6$. Interpret $B^N$ as the open unit ball $\{x\in{\mathbb{R}}^N:|x|<1\}$, and interpret the boundary map $\operatorname{Id}$ as $x\mapsto x$. Define the admissible gradient fields by \[ \mathcal V=\left\{u\in H^1(B^N;{\mathbb{R}}^N):\ \exists\phi\in H^2(B^N;{\mathbb{R}}),\ u=\nabla\phi\ \text{a.e. in }B^N,\ \operatorname{Tr}(\nabla\phi)(x)=x\ \text{on }\partial B^N\right\}, \] where $\operatorname{Tr}$ is the Sobolev trace and the boundary equality is understood in the trace sense. Uniqueness concerns the vector field $u$, not its scalar potential, which can be changed by an additive constant.

For a parameter $\varepsilon$, let the source-designated radial vortex be \[ u_\varepsilon(x)=f_\varepsilon(|x|)\frac{x}{|x|},\qquad x\in B^N\setminus\{0\}. \] The value at the origin does not affect its Sobolev equivalence class; the supplied record does not specify the profile equation, profile regularity, or normalization defining $f_\varepsilon$.

Intended mathematical question. Is $u_\varepsilon$ the unique minimizer of the source-designated Ginzburg--Landau energy $G_\varepsilon$ over $\mathcal V$? Explicitly, the intended assertion is that $u_\varepsilon\in\mathcal V$, that \[ G_\varepsilon(u_\varepsilon)\leq G_\varepsilon(u)\qquad\text{for every }u\in\mathcal V, \] and that \[ u\in\mathcal V,\quad G_\varepsilon(u)=G_\varepsilon(u_\varepsilon) \quad\Longrightarrow\quad u=u_\varepsilon\ \text{a.e. in }B^N. \] The record does not provide the formula or effective domain of $G_\varepsilon$, the allowed range or quantification of $\varepsilon$, or a complete definition of the radial vortex. Thus this expresses the intended minimization and uniqueness question but does not determine a fully specified variational problem. The dimension range above follows the supplied cleaned statement. The accompanying literature assessment instead refers to a question for all $N\geq2$; that broader range is not substituted here.

Short English statement

For each integer dimension $2\leq N\leq6$, the question asks whether the designated radial vortex uniquely minimizes the designated Ginzburg--Landau energy among gradients of real $H^2$ potentials whose gradient equals the identity map on the boundary. The energy, radial profile, and parameter range require clarification. The supplied classification is partially solved; its dated literature assessment has not been independently verified.

Sources

• Supplied local UnsolvedMath problem registry, permanent ID $30005994$, title: Gradient-Constrained Ginzburg--Landau Minimizers; exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. The record gives proposed year 2024 and names no individual proposer.
• Supplied report citation: Calculus of Variations (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/37, pp. 2113--2178. https://doi.org/10.4171/owr/2024/37. This is the authoritative report citation supplied with the record; it has not been independently consulted here.
• Supplied literature source: Ignat--Nahon--Nguyen, gradient fields. https://arxiv.org/abs/2310.11384.
• Supplied literature source: Oberwolfach Report 37/2024. https://ems.press/content/serial-article-files/50045.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that uniqueness over gradient fields is proved for $N\geq4$, while describing the question for all $N\geq2$ as not fully settled. This is supplied dated curation, not an independently verified literature assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim, not verification performed for this draft.