Martinez–Maure Support Functions as Singular Minimizers

Definitions and mathematical statement

Let $R$ denote the function called the support function of the Martínez--Maure example in the supplied record. The record does not give its formula, its domain, or a construction identifying the example and its four cusps unambiguously.

To display the intended variational question, write \[ B_1=\{x\in{\mathbb{R}}^n:|x|<1\},\qquad I_F[u]=\int_{B_1}F(\nabla u(x))\,{\,\mathrm{d}} x. \] Here the ambient dimension $n$, the admissible class of integrands $F$, and the admissible competitor class $\mathcal U_R$ must be supplied from the source. The notation $\mathcal U_R$ stands for the competitors with the intended boundary conditions relative to $R$; those conditions are absent from the record. A function is Lipschitz on $B_1$ if there exists $L<\infty$ such that \[ |R(x)-R(y)|\leq L|x-y|\qquad(x,y\in B_1). \] For a specified competitor class, minimization means \[ R\in\mathcal U_R, \qquad I_F[R]\leq I_F[v]\quad\text{for every }v\in\mathcal U_R. \]

Question, subject to unresolved source definitions. Does there exist an admissible integrand $F$ for which $R$ is a Lipschitz singular minimizer of $I_F$, and for which \[ D_F=\{c_1,c_2,c_3,c_4\}, \] where $c_1,c_2,c_3,c_4$ are exactly the four noncoplanar cusps of the Martínez--Maure example?

The record does not define the singular set $D_F$, the regularity failure meant by a singular minimizer, or the identification of the cusps with points in the domain of $F$. If the cusps are four points in a Euclidean space, their stated noncoplanarity means that they lie in no affine plane; their coordinates are not supplied. No convention for these missing definitions, no convexity or ellipticity hypotheses on $F$, and no boundary conditions are imposed here. Thus the displayed question preserves the supplied target but requires these specifications before it becomes a fully determined mathematical assertion.

Short English statement

The question asks whether the support function of the Martínez--Maure example minimizes an integral depending on its gradient while remaining Lipschitz and singular, with the integrand's singular set consisting exactly of the example's four noncoplanar cusps. The supplied record leaves essential definitions and boundary conditions unspecified, so this is a statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30004193$, source problem code catalogue entry, title: Martinez--Maure Support Functions as Singular Minimizers. No proposer is identified; the supplied proposed year is 2019.
• Supplied report citation: Partial Differential Equations (2020), Oberwolfach Reports, pp. 2033--2097, DOI 10.4171/owr/2019/34. Original Oberwolfach report: https://doi.org/10.4171/owr/2019/34. The citation's 2020 label and its 2019 DOI component are retained as supplied.
• Supplied literature source: Minimizers of Convex Functionals with Small Degeneracy Set, https://arxiv.org/abs/1903.03103.
• Supplied literature source, labeled "Martinez--Maure primary counterexample": https://www.inspe-paris.fr/system/files/2021-06/cras2001.pdf.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and describes this support-function example as a conjectural singular minimizer, reporting that no proof or counterexample to this specific conjecture was found. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.