and which minimizes
among all possible such partitions. Above, denotes the k-th eigenvalue of the Dirichlet-laplacian on (the eigenvalues are counted with multiplicity).
Existence of optimal partitions for this problem in the class of quasi-open sets was proved by Buttazzo and Bucur. For regularity and qualitative studies of the optimal partitions were obtained by Conti, Terracini, and Verzini and Caffarelli, and Lin. Caffarelli and Lin obtained regularity results for the optimal partition and estimates for the asymptotic behavior of the previous sum when . In particular, they conjectured that for the optimal partition
where is the regular hexagon of area 1 in . Roughly speaking this estimate says that, far from , a tiling by regular hexagons of area is asymptotically close to the optimal partition.
In this work we propose a rigorous proof of the equivalence between that optimization problem and a relaxed formulation providing a complete justification of our numerical approach. Based on this method, we performed numerical simulations for and .
As expected, and up to boundary effects, in our numerical experiments, we obtain partitions that are very close to a tiling by regular hexagons in the case . Provided that the conjecture is true, it can be easily proved that the asymptotic optimal partition for is made of unions of pairs of regular hexagons (of measure ). Again our last numerical computations below illustrate this fact.
We were able to run a series of large computations on parallel supercomputers at the Texas Advanced Computing Center. The domain is again the unit square. Periodicity boundary conditions are not used, as the number of cell ( ) is large enough that we expect that the effect of boundary conditions vanishes in the center of the domain. The computations were run on four layers of recursively refined grid of respective dimension , , and . We used a simple projection operator, and the final objective functions. We observe that the solution corresponds to local patches of tiling by regular hexagons, as we would expect from a ``good'' local minimizer.
Our algorithm can easily be adapted to objective function involving higher order eigenvalues of linear combination of eigenvalues of different order. A classical numerical issue in this case comes from the potential non-differentiability of multiple eigenvalues with respect to changes of the density functions.
The figures below represent the optimal partitions obtained with for , and , respectively, using periodic boundary conditions. As explained previously, the optimal partition for is expected to be obtain by a partition made of pairs of regular hexagons. Again, modulo the flattening necessary to achieve periodicity on a unit cell, this is the configuration that we observe. For , we obtain a periodic tiling by non-regular hexagons, which can be proven to be a sub-optimal solution, as a tiling by regular hexagons would lead to a lower energy. This lack of global optimality is most certainly due to the fact that our objective function admits a great deal of local minima, which are difficult to avoid in optimization problems of this size. Additionally difficulty when is that the k-th eigenvalue of an optimal cell is expected to have multiplicity greater than 1 hence is not differentiable.