Incomplete Self-Similar-Tree DtN Uniqueness Claim

Definitions and mathematical statement

The supplied record concerns an unknown scalar Dirichlet-to-Neumann function $\Lambda$ associated with a self-similar tree and displays the functional equation \[ \Lambda(\omega)\cos\omega-\omega\sin\omega = \sum_{i=0}^{p-1}\frac{\mu_i}{\alpha_i} \left(\cos\omega+\frac{\sin\omega}{\omega}\Lambda(\alpha_i\omega)\right). \] Here $\omega$ is the frequency variable, and $p,\mu_i,\alpha_i$ are parameters whose admissible values and geometric meanings are not specified in the supplied statement. The frequency domain, scalar field, tree construction, boundary conditions, and regularity requirements on $\Lambda$ are also unspecified. The displayed expressions require nonzero denominators wherever they are evaluated; no interpretation at $\omega=0$ is supplied.

The requested assertion is that there are at most two "homeomorphic solutions" and that $\Lambda(0)$ uniquely determines a solution. To express only its quantifier structure, let $\mathcal S$ stand for the intended, presently unspecified solution class for fixed admissible tree data and parameters. Provided evaluation at zero is defined for every member, the proposed conclusions would be \[ |\mathcal S|\leq 2, \qquad \forall\Lambda_1,\Lambda_2\in\mathcal S,\quad \Lambda_1(0)=\Lambda_2(0)\Longrightarrow\Lambda_1=\Lambda_2. \] The task is to prove or disprove these conclusions once the missing specifications are recovered. This notation does not define the phrase "homeomorphic solutions": the source gives neither spaces between which the solutions should be homeomorphisms nor another precise meaning for that phrase. Consequently, these formulas record the intended claim but do not constitute a recovered, well-posed conjecture. The supplied statement-verification record marks the formulation as unrecoverable and supplies no clean replacement.

Short English statement

The source asks whether the displayed self-similar-tree Dirichlet-to-Neumann equation has at most two solutions in a class called "homeomorphic," with each solution uniquely determined by its value at zero. That class, the parameter assumptions, the frequency domain, and the interpretation at zero are missing, so a precise conjecture cannot yet be recovered.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $30001423$, source code catalogue entry, titled "Incomplete Self-Similar-Tree DtN Uniqueness Claim." The record belongs to the Oberwolfach Reports Open Problems collection and assigns the proposed year 2010; no individual proposer is supplied.
• Computational Electromagnetism and Acoustics (2010), Oberwolfach Reports, DOI 10.4171/owr/2010/10, pp. 415--519. Supplied source report: https://doi.org/10.4171/owr/2010/10. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Joly--Kachanovska--Semin, 2019, supplied literature reference: https://doi.org/10.3934/nhm.2019010.
• Oberwolfach Report 10/2010, supplied literature reference: https://doi.org/10.4171/owr/2010/10.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem "partially solved" and states that Joly, Kachanovska, and Semin prove uniqueness of the relevant DtN functional equation under appropriate assumptions and develop the fractal-tree theory, while the full homeomorphism and at-most-two formulation is not established in the cited work. This is supplied dated curation derived from the OWR assessment, not an independently verified literature assessment; its status and cited claims require independent verification.
• The supplied statement-verification field marks the statement as "unrecoverable" because "homeomorphic solutions" is undefined for the displayed scalar function as written. No separate clean statement is promoted; the original record is retained publicly for provenance. The supplied record was updated at 2026-09-16T12:06:53Z.