Inf–Sup Stability for Nonconforming Multiple Traces

Definitions and mathematical statement

The supplied record concerns a spectral nonconforming Petrov--Galerkin multiple-traces discretization. Its corrected formulation uses the Cartesian product spaces \[ X_L:=X_L^0\times\cdots\times X_L^M, \qquad Y_L:=Y_L^0\times\cdots\times Y_L^M. \] Here the component spaces, the discretization index $L$, the number of components $M+1$, and their admissible values are not specified. The original extracted formulation instead describes tensor products; the supplied evidence does not resolve this discrepancy.

Writing \[ b_s(\lambda,\phi):=\langle F(s)\lambda,\phi\rangle_\times, \qquad \ell_g(\phi):=\langle g,\phi\rangle_\times, \] the stated discrete problem is to find $\lambda\in X_L$ such that \[ b_s(\lambda,\phi)=\ell_g(\phi) \qquad\text{for every }\phi\in Y_L. \] The operator $F(s)$, its parameter domain, the admissible data $g$, and the pairing $\langle\cdot,\cdot\rangle_\times$ are not defined in the record.

Requested stability statement, conditional on missing specifications. Once the intended nontrivial normed trial and test spaces, operator, and pairing have been specified, the requested trial-side inf--sup estimate would have the form \[ \inf_{0\ne\lambda\in X_L} \sup_{0\ne\phi\in Y_L} \frac{\left|\langle F(s)\lambda,\phi\rangle_\times\right|} {\|\lambda\|_{X_L}\,\|\phi\|_{Y_L}} \geq\beta_L(s)>0. \] The source does not specify the norms, whether the spaces are real or complex, whether the pairing is bilinear or sesquilinear, the admissible parameter values, or whether the lower bound must be uniform in $L$, $s$, or other parameters. Consequently this display records the shape of the requested estimate, not a fully specified conjecture.

The supplied target also asks that the stability condition establish existence and uniqueness of the discrete solution. A positive trial-side inf--sup bound yields at most one solution, but existence for every admissible load requires additional hypotheses ensuring surjectivity of the induced discrete operator. Neither the dimensions of the trial and test spaces nor an appropriate complementary condition is supplied. The intended complete well-posedness assertion therefore remains unresolved.

Short English statement

The problem asks for inf--sup stability and consequent existence and uniqueness for a spectral nonconforming Petrov--Galerkin multiple-traces discretization. Essential spaces, norms, operator definitions, parameter ranges, and solvability hypotheses are missing. The original and corrected records also disagree about tensor versus Cartesian products, so this is a conditional statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30004378$, source problem code catalogue entry, title Inf--Sup Stability for Nonconforming Multiple Traces, in the collection Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2020.
• Supplied report citation: Boundary Element Methods (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/5, pp. 273--376. Source URL and literature-source link, labelled "Oberwolfach report": https://doi.org/10.4171/owr/2020/5. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No exact page or passage for this fragment is supplied.
• Literature assessment dated 2026-08-22: the supplied status is "open", and the supplied assessment describes the extracted statement as a malformed fragment concerning a spectral nonconforming Petrov--Galerkin formulation that does not state a resolved theorem. This is supplied dated curation, not an independently verified literature assessment. The background lists the report above as a source checked and expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and notation repaired for self-contained publication. These are supplied source-verification claims, not verification performed for this draft. They do not resolve the missing mathematical specifications identified here.