Asymptotic hexagonal optimality of optimal partitions for the first Dirichlet eigenvalue (the Caffarelli–Lin conjecture)

For a bounded planar domain $\Omega$ and an integer $m$, partition $\Omega$ into $m$ disjoint subdomains so as to minimize the sum of the first Dirichlet eigenvalues $\lambda_1$ of the cells. Caffarelli and Lin (2007) conjectured that as $m\to\infty$ the optimal partitions converge, away from the boundary, to the regular hexagonal honeycomb — most cells becoming asymptotically congruent regular hexagons of equal area, with deviations confined to a boundary layer of order $\sqrt{m}$ cells — so that the normalized minimal energy satisfies $\lim_{m\to\infty}|\Omega|\mathcal{L}_m(\Omega)/m^2=\lambda_1(H)$, where $H$ is the unit-area regular hexagon. Geng and Lin (2022) proved that this limit exists as a shape-independent constant $c_0$; identifying $c_0$ with $\lambda_1(H)$ and proving the geometric hexagonal convergence remain open. This is the spectral analogue of Hales's honeycomb theorem for perimeter and is proved only for Cheeger and Robin analogues or under convexity and periodicity restrictions.

Background

Let $\Omega\subset\mathbb{R}^2$ be a bounded Lipschitz domain. For a measurable set $\omega$, $\lambda_1(\omega)$ denotes the first eigenvalue of the Dirichlet Laplacian on $\omega$ (the Laplacian with zero boundary condition on $\partial\omega$); it scales as $\lambda_1(t\omega)=t^{-2}\lambda_1(\omega)$. The optimal partition problem of Caffarelli and Lin asks for a partition of $\Omega$ into $m$ mutually disjoint subdomains $\omega_1,\dots,\omega_m$ minimizing

$$\mathcal{L}_m(\Omega)=\inf\Big\{\sum_{i=1}^m\lambda_1(\omega_i):\ \Omega=\bigcup_{i=1}^m\omega_i,\ \omega_i\cap\omega_j=\emptyset\Big\}.$$

They proved that (relaxed) minimizers exist as classical solutions, that the interfaces between cells are smooth outside a singular set of Hausdorff dimension at most $n-2$, and the asymptotic comparability $\mathcal{L}_m(\Omega)\simeq m\lambda_m(\Omega)$ with the $m$-th Dirichlet eigenvalue of $\Omega$.

The Faber–Krahn inequality states that among sets of given area the disk minimizes $\lambda_1$; hence every cell satisfies $\lambda_1(\omega_i)|\omega_i|\ge\lambda_1(D)$ for the unit-area disk $D$, which yields the first-order lower bound $\lambda_1(D)$ on the normalized energy $|\Omega|\mathcal{L}_m(\Omega)/m^2$. Equality would force every cell to be a disk, which is impossible since disks do not tile the plane, so the question is which cell shapes an optimal packing actually uses. For the perimeter functional this is exactly Hales's honeycomb theorem: the regular hexagonal tiling minimizes total perimeter among unit-area partitions of the plane. For the spectral functional, Caffarelli and Lin conjectured the analogous statement: as $m\to\infty$ the optimal partition of a planar domain approaches the regular hexagonal honeycomb, most cells becoming asymptotically congruent regular hexagons of area $|\Omega|/m$, except in a boundary layer of order $\sqrt{m}$ cells near $\partial\Omega$. Numerical evidence has consistently supported this picture since the Fleming–Viot particle simulations of Burdzy, Hołyst, Ingerman and March on the disk (1996) and the dedicated computations of Bourdin, Bucur and Oudet, which show hexagonal interior patterns with roughly $\sqrt{m}$ pentagonal cells hugging the boundary. Writing $H$ for the unit-area regular hexagon, the sharp form of the conjecture is

$$\lim_{m\to\infty}\frac{|\Omega|}{m^2}\mathcal{L}_m(\Omega)=\lambda_1(H).$$

together with local geometric convergence of the minimizers to the hexagonal tiling.

Geng and Lin proved that the limit exists and equals a shape-independent constant $c_0$ for every bounded smooth domain, with $\lambda_1(D)\le c_0\le\lambda_1(H)$; Bourgain and Steinerberger, via a quantitative Faber–Krahn inequality combined with packing properties of disks in the plane, improved the lower bound to $c_0\ge\lambda_1(D)+\varepsilon_0$ for some $\varepsilon_0>0$, so the Faber–Krahn (disk) constant is not attained. For related functionals the honeycomb asymptotics is proved: for the Cheeger constant $h(\omega)=\inf\{\mathrm{Per}(E,\mathbb{R}^2)/|E|:E\subseteq\omega\}$ (for the criterion of minimizing the largest Cheeger constant among the cells) and for the first eigenvalue of the Laplacian under Robin boundary conditions ($\partial_\nu u+\alpha u=0$ on the boundary, with fixed boundary parameter $\alpha$), Bucur and Fragalà established the honeycomb asymptotics without constraints. For $\lambda_1$ itself, the conjecture is proved only under an a priori convexity assumption on the cells (Bucur–Fragalà–Velichkov–Verzini, conditionally on open facts about $\lambda_1$ among convex pentagons, hexagons and heptagons of fixed area) and, more recently, among convex periodic tessellations of the plane (Cesaroni–Fragalà–Novaga, using that $\lambda_1$ decreases under Steiner symmetrization, the rearrangement of a set that makes it symmetric about a line while preserving area). The unconstrained, non-periodic problem for the first Dirichlet eigenvalue — the original Caffarelli–Lin conjecture — remains open.

Problem Statement

Let $\Omega\subset\mathbb{R}^2$ be a bounded Lipschitz domain and let $\mathcal{L}_m(\Omega)=\inf\{\sum_{i=1}^m\lambda_1(\omega_i)\}$ over all partitions $\Omega=\bigcup_{i=1}^m\omega_i$ into $m$ disjoint subdomains, where $\lambda_1$ is the first Dirichlet eigenvalue. It is known (Geng–Lin) that the limit

$$c_0=\lim_{m\to\infty}\frac{|\Omega|}{m^2}\,\mathcal{L}_m(\Omega)$$

exists and is independent of $\Omega$, and that $\lambda_1(D)+\varepsilon_0\le c_0\le\lambda_1(H)$ for the unit-area disk $D$, the unit-area regular hexagon $H$ and some $\varepsilon_0>0$. Prove or refute the Caffarelli–Lin hexagonal packing conjecture, whose two components jointly state that the large-$m$ optimal partitions are asymptotically the regular hexagonal honeycomb:

(i) (sharp constant) $c_0=\lambda_1(H)$: the regular hexagonal tiling is asymptotically optimal for the sum of first Dirichlet eigenvalues among all partitions — with no convexity, polygonality or periodicity restriction on the cells;

(ii) (limiting geometry) for every sequence of optimal $m$-partitions with $m\to\infty$ (up to extraction of subsequences), all but $o(m)$ cells are, up to translation and scaling to common area, asymptotically congruent to the regular hexagon, the exceptional cells being concentrated near $\partial\Omega$ in a boundary layer expected to contain on the order of $\sqrt{m}$ cells.

A refutation may consist of a rigorous construction of admissible partitions of a bounded planar domain whose normalized energy is asymptotically strictly below $\lambda_1(H)$ (contradicting (i), and thereby (ii)), or a proof that optimal partitions fail to converge locally to the hexagonal honeycomb. Both parts together constitute the source question; part (i) alone is the sharp asymptotic constant identification.

References

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