# Optimal partitions

Optimal partitions of the Dirichlet laplacian

During the mini-workshop ‘Shape Analysis for Eigenvalues’ at the Mathematisches Forschungsinstitut Oberwolfach in April 2007, I became interested in the following problem, proposed by G. Buttazzo (Bucur, Buttazzo, & Figueiredo, 1999).

Let $\Omega$ be a domain partitioned in $k$ regions $A_1, \dots, A_k$, and

where $\lambda_1(A_i)$ is the first eigenvalue of the Dirichlet Laplacian on $A_i$. When $\Omega$ and $k$ are ‘large’, L. Cafferelli and F.H. Li conjectured that the partition minimizing $\phi$ consists of a tiling by regular hexagons (Cafferelli & Lin, 2007). With É. Oudet and D. Bucur, I have implemented problem on supercomputers (Bourdin, Bucur, & Oudet, 2009). Figure 1 below represents the partition in subdomains and the eigenfunctions when $k=512$. As expected, the numerical solution consists of regular hexagons away from the boundary of $\Omega$. When optimizing for the sum of the second eigenvalue of the Dirichlet-Laplacian, the optimal partition seems also to consist of regular hexagons, but optimization of the sum of the third eigenvalue leads to more complex geometries.

The implementation of the algorithms described in (Bourdin, Bucur, & Oudet, 2009) is available under an open-source license on bitbucket.

Figure 1: From left to right: Optimal partition of the unit square into 512 regions for the first eigenvalue of the Dirichlet-Laplacian, sum of the associated eigenfunctions, optimal partitions for the sum of the second and third eigenvalues of the Dirichlet Laplacian for $k$ = 8 cells.

### References

1. Bucur, D., Buttazzo, G., & Figueiredo, I. (1999). On the attainable eigenvalues of the Laplace operator. SIAM J. Math. Anal., 30(3), 527–536.
2. Cafferelli, L. A., & Lin, F. H. (2007). An optimal partition problem for eigenvalues. J. Sci. Comput., 31(1-2), 5–18.
3. Bourdin, B., Bucur, D., & Oudet, É. (2009). Optimal Partitions for Eigenvalues. SIAM J. Sci. Comput., 31(6), 4100–4114. DOI:10.1137/090747087 Download

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