Sign-Definite Variational Formulations of Helmholtz Impedance Problems

Definitions and mathematical statement

Source question. Are there other known sign-definite variational formulations of the Helmholtz impedance boundary-value problem?

A variational formulation expresses a boundary-value problem through identities tested against functions in specified function spaces. The supplied record does not specify the Helmholtz equation and its sign convention, the spatial domain or its regularity, the impedance boundary condition and its coefficients, the admissible data, the trial and test spaces, or the intended meaning of sign-definiteness. These choices are therefore left unresolved. In particular, no particular positivity, coercivity, or parameter-uniform estimate is asserted here. The word "other" refers to formulations that are not reproduced in the supplied record, so the comparison class is also unspecified.

Qualification of the intended target. The supplied statement-verification note describes this question as an expository transition in the report, immediately followed by examples and a general transformation, rather than a standalone open problem. Those examples and that transformation are not included in the supplied record. Accordingly, this draft preserves the source question but cannot extract a uniquely specified mathematical proposition or set of quantifiers. The registry labels the statement unrecoverable; its separate open status is supplied curation, not a verified mathematical conclusion.

Short English statement

The source asks about additional sign-definite variational formulations of a Helmholtz problem with an impedance boundary condition. The governing assumptions, the meaning of sign-definiteness, and the existing formulations used for comparison are unspecified. The supplied verification note treats the sentence as an expository transition, so it does not determine a standalone open problem.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002279$, exact source problem code catalogue entry; title: Sign-Definite Variational Formulations of Helmholtz Impedance Problems. Collection: Oberwolfach Reports Open Problems. The record gives the proposed year as 2013 and supplies no individual proposer.
• Computational Electromagnetism and Acoustics (2013), Oberwolfach Reports, pp. 129--237, DOI: 10.4171/owr/2013/03. Supplied report citation and literature-source link labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2013/03. This reference has not been independently visited or verified for this draft.
• Supplied literature assessment, checked 2026-08-21: status open; described as an open-ended request for alternative sign-definite formulations without a uniquely specified proposition or complete canonical answer. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background requests independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note: the report immediately provides examples and a general transformation after the question; the extraction is classified as an expository transition, no separate clean statement is promoted, and the original record is retained for provenance.