Existence of Optimal Two-Phase Eigenvalue Configurations

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{R}}^d$ be bounded, and let $0<m<|\Omega|$ and $0<\alpha<\beta$. Interpret $|\cdot|$ as Lebesgue measure. Define the admissible class \[ \mathcal B_m(\Omega)=\{B\subset\Omega:B\text{ is Lebesgue measurable and }|B|=m\}. \] For $B\in\mathcal B_m(\Omega)$, put $A=\Omega\setminus B$ and \[ \sigma_B(x)=\alpha\chi_A(x)+\beta\chi_B(x),\qquad x\in\Omega, \] where $\chi_E$ denotes the characteristic function of $E$.

The supplied record denotes by $\lambda(B)$ the first Dirichlet eigenvalue associated with this two-phase configuration. It does not specify the differential operator, the eigenvalue equation, or the variational definition connecting $\sigma_B$ to $\lambda(B)$. Consequently, $\lambda(B)$ cannot be defined unambiguously from this record alone. Moreover, only boundedness of $\Omega$ is stated: its measurability is implicit in the volume notation, and openness or another framework for the Dirichlet problem is not specified.

Subject to resolving these missing specifications, the intended existence question is: for every admissible choice of $d,\Omega,m,\alpha,\beta$ satisfying the supplied hypotheses, does there exist $B_*\in\mathcal B_m(\Omega)$ such that \[ \lambda(B_*)=\inf_{B\in\mathcal B_m(\Omega)}\lambda(B)? \] The optimization is over all measurable configurations with the prescribed volume, with no further geometric restriction stated. This is a conditional formulation of the extracted question; the missing operator and domain hypotheses remain unresolved.

Short English statement

For permanent UnsolvedMath ID $30002244$, the question asks whether the least first Dirichlet eigenvalue among all measurable arrangements of two fixed phases with prescribed volumes is attained. The supplied record leaves the eigenvalue operator and the precise domain assumptions unspecified. Its dated curation labels the existence question open and separately reports that a conjectured optimizer is false in general; that report does not settle the existence question.

Sources

• Local UnsolvedMath registry: supplied complete source record for permanent ID $30002244$, entitled Existence of Optimal Two-Phase Eigenvalue Configurations, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as $2012$; no individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Geometries, Shapes and Topologies in PDE-based Applications (2013), Oberwolfach Reports, pp. 3375--3415, DOI 10.4171/owr/2012/57. Supplied link: https://doi.org/10.4171/owr/2012/57.
• Supplied literature source, labeled "Oberwolfach report": https://ems.press/content/serial-article-files/46428.
• Supplied literature source, labeled "low contrast counterexample paper": https://epubs.siam.org/doi/10.1137/110847822.
• Supplied literature source, labeled "global minimizer paper": https://www.numdam.org/item/COCV_2014__20_2_362_0.pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports that the conjectured optimizer is false in general, and separately states that existence for the unrestricted two-phase problem remains open. It explicitly distinguishes the failed optimizer conjecture from the existence question. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment.
• The registry marks the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. These are supplied verification claims, not verification performed for this draft.