Collapse of Multivalued Minimizers with Single-Valued Boundary Data

Definitions and mathematical statement

For a positive integer $Q$ and a target set $X$, write \[ \mathcal A_Q(X)=\left\{\sum_{i=1}^{Q}\delta_{x_i}:x_i\in X\right\}, \qquad D_Q(x)=Q\delta_x, \] where the sums are formal unordered collections with multiplicity, and $\delta_x$ denotes one copy of $x$. Thus $D_Q(x)$ is the completely collapsed $Q$-valued point. This notation replaces the source's double-bracket notation. For a single-valued map $v$, $D_Q\circ v$ denotes its pointwise collapsed $Q$-valued counterpart.

The source asks the following question. Suppose that $u:B\to\mathcal A_Q(X)$ minimizes a quasiconvex functional in the intended admissible class and has boundary data \[ u|_{\partial B}=D_Q\circ\phi, \] where $\phi:\partial B\to X$ is single-valued. Must there exist a single-valued minimizer $\Phi:B\to X$ for the corresponding single-valued variational problem, with boundary data $\Phi|_{\partial B}=\phi$, such that \[ u=D_Q\circ\Phi\quad\text{in }B? \] The quantifier concerns the given minimizer $u$: the question asks whether every minimizer satisfying the stated collapsed boundary condition must itself collapse in this way, rather than merely whether a collapsed minimizer exists.

This is a schematic formulation requiring additional hypotheses before it defines a precise variational problem. The supplied record does not specify the domain $B$, target $X$, dimensions, admissible function spaces, boundary trace convention, energy density, meaning of quasiconvexity for the multivalued functional, or relation to the single-valued functional. The equality in $B$ must be interpreted in the eventual admissible function space; its pointwise or almost-everywhere meaning is not specified. The notation above supplies no missing analytic structure or energy assumptions.

Short English statement

If a multivalued energy minimizer has all $Q$ sheets equal on the boundary, must all its sheets coincide throughout the domain with a single-valued minimizer? The supplied record leaves the variational setting and the associated single-valued problem unspecified, so this remains a statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002993$, title: Collapse of Multivalued Minimizers with Single-Valued Boundary Data; exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied source citation: Mini-Workshop: Scales in Plasticity (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/53, pp. 3007--3048. https://doi.org/10.4171/owr/2015/53. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled Oberwolfach Q-valued report: https://ems.press/content/serial-article-files/46600.
• Supplied literature source, labeled Q-valued Dirichlet theory: https://arxiv.org/abs/0803.0060.
• Supplied literature source, labeled Current Q-graph framework: https://arxiv.org/abs/2603.20788.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the general branching implication for minimizers with collapsed multiple-of-one-valued boundary data remains unresolved, and that existing Dirichlet theory does not establish the claimed conclusion for the full variational setting. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The source itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, but its accompanying verification describes only safe notation, typography, and TeX-delimiter cleanup, with hypotheses, quantifiers, and subquestions preserved. This label does not establish mathematical correctness or verification of the cited sources.