Fractal Optimizers in Spectral Isoperimetric Problems

Definitions and mathematical statement

An eigenvalue isoperimetric problem is understood here only schematically as an optimization problem for a quantity determined by eigenvalues, subject to geometric constraints. For a specified admissible class $\mathcal A$ and a specified objective $J\colon\mathcal A\to{\mathbb{R}}\cup\{+\infty\}$, an object $X_*\in\mathcal A$ is a global minimizer if \[ J(X_*)=\inf_{X\in\mathcal A}J(X). \] This notation describes the minimizer terminology only; the supplied record does not specify $\mathcal A$, $J$, or the underlying eigenvalue problem.

The source question is: Can an isoperimetric problem for eigenvalues have a fractal minimizer or other fractal solution?

The question asks whether such a phenomenon can occur, without specifying a particular optimization problem. A precise instance would require an ambient space, an operator and its realization, boundary conditions where applicable, an eigenvalue objective, geometric constraints, an admissible class, and a definition of fractality. In particular, the record does not say whether fractality concerns the optimizing object itself, its boundary, or another associated set. The phrase "other fractal solution" also has no specified mathematical meaning. Consequently, the displayed minimization formula is only a schematic description of one branch of the question; a fully quantified mathematical statement cannot be recovered from the supplied record without additional choices.

Short English statement

Can an eigenvalue optimization problem subject to geometric constraints admit a fractal minimizer, or another kind of fractal solution? The source leaves the operator, constraints, admissible objects, meaning of fractality, and alternative notion of solution unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001160$, title "Fractal Optimizers in Spectral Isoperimetric Problems," exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives the proposed year as 2009. No individual proposer is supplied.
• Low Eigenvalues of Laplace and Schrödinger Operators (2009), Oberwolfach Reports, pp. 355--428, DOI 10.4171/owr/2009/06. Supplied source-report and literature-source link: https://doi.org/10.4171/owr/2009/06. The collection describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "regularity of optimal spectral domains": https://doi.org/10.1007/s00205-014-0801-6. No authors, title beyond this descriptive label, or relevant theorem are supplied in the record.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and describes existence of a fractal eigenvalue optimizer as an unresolved regularity and attainment question. It states that later spectral-shape theory provides regularity for important relaxed problems without generally resolving this question. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry marks the statement as "exact" and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized. This is a supplied source-verification assertion, not verification performed for this draft.