Laplacians on Nonsymmetric Self-Similar Sets

Definitions and mathematical statement

Let $\mathscr C$ denote the intended class of connected nonsymmetric self-similar sets in the supplied source question. Connectedness means that a set, with its intended topology, cannot be expressed as the union of two disjoint nonempty relatively open subsets. The supplied record does not specify the ambient category, the self-similarity assumptions, or the symmetry condition whose failure defines "nonsymmetric." These parts of $\mathscr C$ therefore remain unresolved.

For $K\in\mathscr C$, let $\mathscr L(K)$ provisionally denote the class of operators that qualify as Laplacians on $K$ in the intended source construction. This is a placeholder for an unspecified analytic definition, not a definition of an admissible Laplacian.

The source question has the quantifier structure \[ \text{Is it true that }\quad \forall K\in\mathscr C,\qquad \mathscr L(K)\ne\varnothing\,? \] Equivalently, does every connected nonsymmetric self-similar set admit a Laplacian in the intended sense?

A fully specified mathematical existence statement cannot be recovered from this record alone. In particular, the record does not state the admissible operators, their function spaces or domains, a reference measure if required, boundary conditions if applicable, or compatibility requirements with the self-similar structure. It also does not identify the construction to which the supplied verification note refers. These omissions must be resolved before the displayed schema can be treated as a precise analytic assertion. No additional restrictions on $K$, and no analytic definition of Laplacian, are imposed here.

Short English statement

The question asks whether every connected nonsymmetric self-similar set supports a Laplacian of the kind intended by the source. The supplied record leaves both the class of sets and the required Laplacian insufficiently specified, so this draft preserves the universal existence question while marking the missing definitions. Its supplied status is open; that status has not been independently verified.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30002164$, title "Laplacians on Nonsymmetric Self-Similar Sets," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2012$ and names no proposer.
• Supplied source citation: Optimal and Near Optimal Configurations on Lattices and Manifolds (2013), Oberwolfach Reports, pp. 2429--2492, DOI 10.4171/owr/2012/40. https://doi.org/10.4171/owr/2012/40. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions. No exact problem page is supplied.
• Supplied literature reference labeled "OWR report 40": https://doi.org/10.4171/owr/2012/40.
• Supplied literature reference labeled "Kigami--Strichartz": https://doi.org/10.1016/j.chaos.2011.05.003. The record supplies no further bibliographic details for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that Laplacians have been constructed for important classes such as post-critically finite (pcf) self-similar sets, while no general existence theorem covers arbitrary connected nonsymmetric self-similar sets. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the status and cited claims require independent verification.
• The registry assigns statement status corrected_verified and reports that the official report asks whether a Laplacian construction extends to every connected nonsymmetric self-similar set. This verification claim is supplied metadata; neither linked source has been independently consulted for this draft.