Optimal spectral partitions for the sum and the maximum of the first Dirichlet eigenvalues on the square, the disk, and the equilateral triangle

For a domain $\Omega$ partitioned into $k$ open cells, compare two spectral energies: the maximum $\mathfrak{L}_k(\Omega)$ and the average $\mathfrak{L}_{k,1}(\Omega)$ of the first Dirichlet eigenvalue $\lambda_1$ over the cells. Bogosel and Bonnaillie-Noël (2017) computed candidate optimal $k$-partitions of the unit square, the unit disk, and the equilateral triangle for both energies, proved through an $L^2$-mass criterion that most of the max-candidates cannot minimize the sum, and built better curved-interface candidates for the sum — while leaving every decisive question open: no candidate's global optimality is proved, the square with $k=7$ and the equilateral triangle with $k=4$ are numerically ambiguous for the sum, the triangle cases $k\in\{6,10\}$ are undecided by the criterion, and the circular-arc structure of the curved candidates has no analytic justification. The problem is to rigorously determine the optimal partitions and energies for both functionals on these three benchmark domains.

Background

Partitioning a planar domain so as to minimize spectral quantities of the pieces is a classical topic between spectral geometry, shape optimization, and the calculus of variations, motivated among other things by limits of competing-species models in population dynamics and by the honeycomb conjecture (the statement, proved by Hales, that regular hexagons tile the plane into equal-area cells with least total perimeter). For an open set $D\subset\mathbb{R}^2$, let $\lambda_1(D)$ denote the first eigenvalue of the Dirichlet Laplacian on $D$, i.e. the square of the fundamental frequency of a membrane fixed along the boundary of $D$. A $k$-partition $\mathcal{D}=(D_1,\dots,D_k)$ of a domain $\Omega$ is a family of $k$ mutually disjoint, open, connected sets whose union covers $\Omega$ up to a set of measure zero (the interfaces between cells). Two energies are compared: the maximum functional $\mathfrak{L}_k(\Omega)=\inf_{\mathcal{D}}\max_{1\le i\le k}\lambda_1(D_i)$ and the sum (equivalently, average) functional $\mathfrak{L}_{k,1}(\Omega)=\inf_{\mathcal{D}}\frac{1}{k}\sum_{i=1}^{k}\lambda_1(D_i)$. Partitions attaining or approaching these infima are called spectral minimal partitions.

The general theory is well developed. Minimizers exist for both functionals and are strongly regular away from finitely many singular points: interfaces are analytic arcs meeting three at a time at equal angles of $2\pi/3$ (the honeycomb-type junction condition), with an analogous equal-angle structure at boundary singular points (Helffer, Hoffmann-Ostenhof, and Terracini, 2009; sharpened for broad classes of spectral partition costs by Tavares and Zilio, 2021, and up to the fixed boundary by Ognibene and Velichkov, 2024). A partition is called bipartite if adjacent cells can be colored with two colors; for the maximum functional, bipartite minimal partitions are exactly nodal partitions, i.e. the connected components of the set where a Laplacian eigenfunction keeps a constant sign. Consequently $\mathfrak{L}_k(\Omega)=\lambda_k(\Omega)$ holds precisely when the $k$-th eigenvalue is Courant-sharp, meaning some $k$-th eigenfunction has exactly $k$ nodal domains; this is essentially the only general mechanism that identifies minimal partitions explicitly. By Courant-sharpness, the maximum functional is settled for the nodal cases $k\in\{1,2,4\}$ on all three of the domains below — in particular the unit square with $k=4$ (four equal subsquares) and the equilateral triangle with $k=4$ (four equilateral sub-triangles; Bérard and Helffer, 2016). For the sum functional no identification theorem exists at all: a nodal partition can be strictly improved.

The source paper (Bogosel and Bonnaillie-Noël, arXiv:1702.01567, 2017) compared the two energies on the three benchmark domains — the square of sidelength 1, the disk of radius 1, and the equilateral triangle of sidelength 1 — for $2\le k\le 10$. It used numerical algorithms of three kinds: a relaxed, penalized iterative method in the spirit of Bourdin, Bucur, and Oudet (2009); a mixed Dirichlet–Neumann eigenproblem ansatz for symmetric candidates, in which symmetry is exploited by solving an eigenproblem on a fundamental subdomain with reflecting (Neumann) conditions along the symmetry lines; and explicit one-parameter families of partitions bounded by circular arcs. It proposed candidates for the maximum, applied a criterion of Helffer and Hoffmann-Ostenhof (2010) — a max-candidate that is the nodal partition of an eigenfunction $\varphi$ cannot minimize the sum if two neighboring cells carry different $L^2$-masses $\int_{D_i}|\varphi|^2$ — to show that most max-candidates cannot be sum-optimal, and then exhibited better sum-candidates with curved interfaces. The $L^2$-mass criterion is inconclusive for partitions built of congruent cells (all masses equal), and on the equilateral triangle also for $k\in\{6,10\}$, where the computed masses of adjacent cells agree to the accessible numerical precision. Its journal companion (Bogosel and Bonnaillie-Noël, Interfaces Free Boundaries, 2018) interpolates the two energies through $p$-norms of the eigenvalues ($p=1$ is the sum, $p=\infty$ the maximum) and formulates the resulting candidates as explicit conjectures, none of them proved.

Problem Statement

For the square of sidelength 1, the disk of radius 1, and the equilateral triangle of sidelength 1, and for $2\le k\le 10$ (the range studied in the source), rigorously determine the minimal $k$-partitions and the minimal energies for the maximum functional $\mathfrak{L}_k(\Omega)=\inf_{\mathcal{D}}\max_{1\le i\le k}\lambda_1(D_i)$ and the sum functional $\mathfrak{L}_{k,1}(\Omega)=\inf_{\mathcal{D}}\frac{1}{k}\sum_{i=1}^{k}\lambda_1(D_i)$, where the infima run over $k$-partitions as defined above. The concrete questions left open by the source are:

(1) Global optimality of the proposed candidates. Prove or disprove that the $k$ equal angular sectors minimize the energies on the disk for $k\in\{2,3,4,5\}$ (the case $k=3$ is the Mercedes-star conjecture; for $k\in\{2,4\}$ the maximum side is already settled by Courant-sharpness, so the open content there is the sum side), that the four subsquares minimize both energies on the square for $k=4$ (and that the minimal 2-partitions of the square, among them two equal rectangles, also minimize the sum for $k=2$), and that the triangular-number partitions ($k=n(n+1)/2$, i.e. $k\in\{3,6,10\}$ in this range) minimize the energies on the equilateral triangle; and prove or disprove optimality of the numerical candidates for the maximum and for the sum on the square, disk, and triangle for the remaining $k$, where only numerical upper bounds are known.

(2) The undecided cases. (a) Square, $k=7$: two near-degenerate candidate partitions have numerically indistinguishable sum-energies — decide which, if either, is optimal, or find the true sum-minimizer. (b) Equilateral triangle, $k=4$: the max-minimizer is the partition into four equilateral sub-triangles, but a less symmetric partition has numerically smaller sum-energy — determine the sum-minimizer and prove it. (c) Equilateral triangle, $k\in\{6,10\}$: the $L^2$-mass criterion does not apply and numerics suggest the sum- and max-minimizers coincide — prove or disprove this coincidence.

(3) Structure of the curved candidates. In the symmetric regimes (disk $k\in\{6,8,9\}$, square $k\in\{3,5\}$) the improved sum-candidates are built from circular-arc interfaces meeting at angles $2\pi/3$; it is not known whether optimal interfaces in these regimes are circular arcs at all, and the one-parameter arc families have no analytic existence, uniqueness, or optimality justification. Resolve this structure question. Each of (1)–(3) bundles several separately refutable claims; the unit of resolution is the single claim, so settling one claim of a bundle (for instance only the disk $k=3$ Mercedes-star case within (1)) is a self-contained result that need not extend to the rest of its sub-question, and a complete solution settles every claim in (1)–(3). Overlapping claims are resolved together: the triangle $k\in\{6,10\}$ coincidence in (2)(c) is implied by the triangular-number claim in (1).

References

1. Beniamin Bogosel and Virginie Bonnaillie-Noël, “Optimal partitions for the sum and the maximum of eigenvalues” (2017). arXiv:1702.01567, DOI 10.48550/arXiv.1702.01567. https://arxiv.org/abs/1702.01567
2. Beniamin Bogosel and Virginie Bonnaillie-Noël, “Minimal partitions for p-norms of eigenvalues”, Interfaces and Free Boundaries 20(1):129–163 (2018). DOI 10.4171/ifb/399. https://doi.org/10.4171/ifb/399 (preprint arXiv:1612.07296, https://arxiv.org/abs/1612.07296)
3. B. Helffer and T. Hoffmann-Ostenhof, “Remarks on two notions of spectral minimal partitions”, Advances in Mathematical Sciences and Applications 20(1):249–263 (2010). Publisher Gakkōtosho, Tokyo; no DOI assigned. Zbl 1222.35067, https://zbmath.org/1222.35067
4. Bernard Helffer, Thomas Hoffmann-Ostenhof, and Susanna Terracini, “Nodal domains and spectral minimal partitions”, Annales de l'Institut Henri Poincaré C, Analyse non linéaire 26(1):101–138 (2009). DOI 10.1016/j.anihpc.2007.07.004. https://doi.org/10.1016/j.anihpc.2007.07.004
5. Blaise Bourdin, Dorin Bucur, and Édouard Oudet, “Optimal partitions for eigenvalues”, SIAM Journal on Scientific Computing 31(6):4100–4114 (2009). DOI 10.1137/090747087. https://doi.org/10.1137/090747087
6. Pierre Bérard and Bernard Helffer, “Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle”, Letters in Mathematical Physics 106(12):1729–1789 (2016). DOI 10.1007/s11005-016-0819-9. https://doi.org/10.1007/s11005-016-0819-9
7. Virginie Bonnaillie-Noël, Bernard Helffer, and Thomas Hoffmann-Ostenhof, “Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators” (2017). arXiv:1711.01174. https://arxiv.org/abs/1711.01174
8. Gregory Berkolaiko, Yaiza Canzani, Graham Cox, Peter Kuchment, and Jeremy L. Marzuola, “Stability of spectral partitions with corners”, Journal of the European Mathematical Society (JEMS), online first (2026). arXiv:2411.02378, DOI 10.4171/jems/1759. https://arxiv.org/abs/2411.02378
9. Gregory Berkolaiko, Yaiza Canzani, Graham Cox, and Jeremy L. Marzuola, “Stability of spectral partitions and the Dirichlet-to-Neumann map”, Calculus of Variations and Partial Differential Equations 61(6), article 203 (2022). DOI 10.1007/s00526-022-02311-7. https://arxiv.org/abs/2201.00773
10. Hugo Tavares and Alessandro Zilio, “Regularity of all minimizers of a class of spectral partition problems”, Mathematics in Engineering 3(1):1–31 (2021). DOI 10.3934/mine.2021002. https://arxiv.org/abs/2002.04064
11. Roberto Ognibene and Bozhidar Velichkov, “Boundary regularity of the free interface in spectral optimal partition problems” (2024). arXiv:2404.05698. https://arxiv.org/abs/2404.05698
12. Roberto Ognibene and Bozhidar Velichkov, “A survey on the optimal partition problem”, La Matematica 5(1), article 6 (2026). DOI 10.1007/s44007-025-00176-8. https://doi.org/10.1007/s44007-025-00176-8 (preprint arXiv:2510.08241, https://arxiv.org/abs/2510.08241)
13. Kwunlun Chu and Shingyu Leung, “A level set method for the Dirichlet k-partition problem”, Journal of Scientific Computing 86(1), article 11 (2021). DOI 10.1007/s10915-020-01368-w. https://doi.org/10.1007/s10915-020-01368-w
14. Dong Wang and Braxton Osting, “A diffusion generated method for computing Dirichlet partitions”, Journal of Computational and Applied Mathematics 351:302–316 (2019). DOI 10.1016/j.cam.2018.11.015. https://doi.org/10.1016/j.cam.2018.11.015
15. O. Cybulski, V. Babin, and R. Hołyst, “Minimization of the Renyi entropy production in the space-partitioning process”, Physical Review E 71(4):046130 (2005). DOI 10.1103/PhysRevE.71.046130. https://doi.org/10.1103/PhysRevE.71.046130
16. Beniamin Bogosel and Dorin Bucur, “Polygonal Faber-Krahn inequality: local minimality via validated computing” (2024). arXiv:2406.11575. https://arxiv.org/abs/2406.11575
17. Beniamin Bogosel, “Efficient algorithm for large spectral partitions” (2017). arXiv:1705.08739. https://arxiv.org/abs/1705.08739