Continuity of Gradient Distance for Monotone Equations

Definitions and mathematical statement

A continuous vector field $G\colon{\mathbb{R}}^2\to{\mathbb{R}}^2$ is strictly monotone if \[ \left(G(p)-G(q)\right)\cdot(p-q)>0 \qquad\text{for every }p,q\in{\mathbb{R}}^2\text{ with }p\ne q. \] The supplied conjecture concerns a Lipschitz solution of $\operatorname{div}G(\nabla u)=0$. The spatial domain and the meaning of solution are not specified. To display a possible interior weak formulation, let $\Omega\subseteq{\mathbb{R}}^2$ be open and let $u\colon\Omega\to{\mathbb{R}}$ be Lipschitz, with \[ \int_\Omega G(\nabla u(x))\cdot\nabla\varphi(x)\,{\,\mathrm{d}} x=0 \qquad\text{for every }\varphi\in C_c^\infty(\Omega). \] This choice of domain and solution concept is an explicit interpretation requiring confirmation; the record supplies no boundary conditions.

The record describes $\mathcal B\subseteq{\mathbb{R}}^2$ as the set where ellipticity degenerates simultaneously from below and above, but supplies no mathematical definition of this set. For a specified set $\mathcal B$, write \[ \operatorname{dist}(p,\mathcal B)=\inf_{b\in\mathcal B}|p-b|. \] With $\inf\varnothing=+\infty$, an ordinary real-valued continuity assertion requires $\mathcal B\ne\varnothing$, or a separately stated convention for the empty case.

Subject to specifying $\mathcal B$ and resolving these qualifications, the supplied conjecture asks whether, for every continuous strictly monotone $G$ and every Lipschitz solution $u$ as above, the almost-everywhere defined function \[ x\longmapsto\operatorname{dist}(\nabla u(x),\mathcal B) \] has a continuous representative. In the nonempty case this interpretation means that there exists $f\in C^0(\Omega;{\mathbb{R}})$ such that \[ f(x)=\operatorname{dist}(\nabla u(x),\mathcal B) \qquad\text{for almost every }x\in\Omega. \] The source does not specify whether this representative interpretation or a stronger pointwise assertion is intended. Without the definition of $\mathcal B$, this remains a conditional formulation of the supplied conjecture, not a fully specified mathematical statement.

Short English statement

The conjecture asks whether the distance of the gradient of a planar Lipschitz solution from a simultaneous ellipticity-degeneracy set varies continuously. The definition of that set, the spatial domain, the solution concept, and the intended interpretation of continuity require clarification.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005995$, source problem code catalogue entry; title: Continuity of Gradient Distance for Monotone Equations. The supplied record attributes the conjecture to an Oberwolfach workshop report 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. The record lists proposed year 2024; the supplied title year 2025 and report identifier 37/2024 are retained without reconciliation.
• 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 open and reports that no primary proof or counterexample was located. This is supplied dated curation, not an independently verified literature assessment.
• The record labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied verification claims; no source visit or independent verification is asserted here.