Equality Cases for Universal Eigenvalue Mean Inequalities

Definitions and mathematical statement

Let $n$ denote the dimension parameter in the supplied formula, and let $E_1\leq E_2\leq\cdots$ be the Dirichlet eigenvalues associated with a domain $\Omega$. The supplied record does not specify the operator, the ambient space, the admissible class of domains, or hypotheses ensuring a discrete spectrum and well-defined real powers of the eigenvalues. The statement below is therefore conditional on the intended spectral setting, which requires clarification from the cited report.

For $p>0$ and a positive integer $J$, define \[ M_p(J)=\left(\frac{n+2p}{nJ}\sum_{j=1}^{J}E_j^p\right)^{1/p}. \] The corrected formulation supplied in the record gives the inequality \[ M_1(J)^2-M_2(J)^2\geq\frac{(E_{J+1}-E_J)^2}{4}. \] Its equality question is: does there exist an admissible domain $\Omega$ and a finite positive integer $J$ such that \[ M_1(J)^2-M_2(J)^2=\frac{(E_{J+1}-E_J)^2}{4}? \] The admissibility conditions and whether $n$ is fixed in advance remain unspecified in the supplied record.

The original extraction instead displayed $M_{1/2}(J)-M_2(J)$ on the left. The supplied verification explicitly identifies this as a transcription error and reports that the official report uses $M_1(J)^2-M_2(J)^2$. This draft follows that supplied correction without claiming independent source verification. The original extraction also mentions $M_0(J)$, described as the corresponding geometric mean, without an explicit normalization; it is not needed for the corrected equality question.

Short English statement

The question asks whether the supplied universal inequality between normalized Dirichlet eigenvalue means has an exact equality case at a finite index. The draft follows the supplied correction to the squared means. The intended operator, domain hypotheses, and dimension quantifier still require clarification. The record labels the question open; that status is supplied dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30001163$, source problem code catalogue entry, title: Equality Cases for Universal Eigenvalue Mean Inequalities. The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives the proposed year as 2009; no individual proposer is identified.
• Low Eigenvalues of Laplace and Schrödinger Operators (2009), Oberwolfach Reports, pp. 355--428, DOI 10.4171/owr/2009/06. Supplied source and literature reference: https://doi.org/10.4171/owr/2009/06. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Harrell--Stubbe, universal bounds, as labeled in the supplied literature sources: https://arxiv.org/abs/0808.1133. No fuller bibliographic description is supplied.
• Supplied statement verification: the record labels the statement corrected_verified and reports that comparison with the official report corrected the left side from $M_{1/2}(J)-M_2(J)$ to $M_1(J)^2-M_2(J)^2$. This verification is reported from the record, not independently performed here.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open, describes the trace-identity inequality as known and asymptotically sharp, and reports that no domain and index giving exact finite-index saturation was found. This is supplied dated curation derived from the OWR record, not independently verified; the status and cited mathematical claims require independent checking.