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 be a domain partitioned in regions , and
where is the first eigenvalue of the Dirichlet Laplacian on . When and are ‘large’, L. Cafferelli and F.H. Li conjectured that the partition minimizing 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 . As expected, the numerical solution consists of regular hexagons away from the boundary of . 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.
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 = 8 cells.
- The TopOpt group at the Technical University of Denmark.
- The Shape and Topology Optimization Group at CMAP (École Polytechnique, France).
- The International Society for Structural and Multidisciplinary Optimization.
- The Journal of Structural and Multidisciplinary Optimization.
- Bucur, D., Buttazzo, G., & Figueiredo, I. (1999). On the attainable eigenvalues of the Laplace operator. SIAM J. Math. Anal., 30(3), 527–536.
- Cafferelli, L. A., & Lin, F. H. (2007). An optimal partition problem for eigenvalues. J. Sci. Comput., 31(1-2), 5–18.
- Bourdin, B., Bucur, D., & Oudet, É. (2009). Optimal Partitions for Eigenvalues. SIAM J. Sci. Comput., 31(6), 4100–4114. DOI:10.1137/090747087 Download