Universal Lower Bounds for Biharmonic Steklov Eigenvalues

Definitions and mathematical statement

Let $n$ denote the ambient Euclidean dimension, whose permitted range is not specified in the supplied record. For a measurable set $\Omega\subset{\mathbb{R}}^n$, write $|\Omega|$ for its $n$-dimensional Lebesgue measure. A set is convex if it contains the line segment joining any two of its points.

The supplied title describes $\rho_k(\Omega)$ as biharmonic Steklov eigenvalues, but the record does not define the underlying differential operator, boundary conditions, the role of the positive parameter $\alpha$, or the eigenvalue indexing convention. Consequently, $\rho_k(\Omega)$ must remain an undefined source symbol here; a complete spectral definition cannot be reconstructed from this record alone.

The conjecture as supplied has the following conditional formulation. For every fixed $\alpha>0$ and $M>0$, there exist constants $a=a(\alpha,M,n)>0$ and $b=b(\alpha,M,n)>0$, independent of $k$ and $\Omega$, such that \[ \begin{aligned} \rho_k(\Omega)&\geq a\,k^{2/(2n-1)} &&\text{for every admissible convex }\Omega\subset{\mathbb{R}}^n\text{ with }|\Omega|=M,\\ \rho_k(\Omega)&\geq b\,k^{1/n} &&\text{for every admissible general }\Omega\subset{\mathbb{R}}^n\text{ with }|\Omega|=M, \end{aligned} \qquad\text{for all }k\in{\mathbb{N}}. \] Here ${\mathbb{N}}$ reproduces the source's index set; its starting index is unspecified. The word "admissible" records missing hypotheses rather than supplying them: the record does not specify boundedness, boundary regularity, connectedness, or the conditions needed to define the spectrum. In particular, "general domain" cannot yet be given a precise class. These omissions must be resolved before the displayed conjecture becomes a fully defined mathematical statement.

The source motivates the conjecture by results for rectangles and unions of rectangles attributed to its reference [5], whose bibliographic identity is absent. The subsequent passage headed "Daniel Grieser: Behavior of the spectrum under degeneration" discusses a different, broader class of problems and does not supply a definition of $\rho_k$ or boundary conditions for this conjecture.

Short English statement

The supplied conjecture asks for uniform lower bounds on $\rho_k(\Omega)$ at fixed volume: growth of order $k^{2/(2n-1)}$ for convex domains and $k^{1/n}$ for general domains, with positive constants depending only on $\alpha,M,n$. The spectral problem and admissible domains are not defined in the record, so this remains a conditional statement requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005616$, source problem code catalogue entry, title "Universal Lower Bounds for Biharmonic Steklov Eigenvalues." No proposer is identified in the record.
• Supplied report citation: Geometric Spectral Theory (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/36, pp. 2017--2106. Supplied source URL: https://doi.org/10.4171/owr/2023/36. The literature source labeled "original OWR report" supplies this same link.
• The original excerpt cites reference [5] for results on rectangles and unions of rectangles. Its authors, title, and bibliographic details are not supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and describes it as a duplicate formulation remaining unresolved beyond the motivating special families. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim, not verification performed for this draft. The registry gives proposed year 2023, while the supplied report citation uses 2024.
• The adjacent excerpt attributes "Behavior of the spectrum under degeneration" to Daniel Grieser. It discusses families of self-adjoint elliptic operators with discrete spectrum, including domain degeneration and semiclassical examples; it does not establish authorship of the preceding conjecture.