Abstract

We study the problem of partitioning domains by minimising functionals based on Robin Laplacian eigenvalues of the partition cells, analogous to the “classical” spectral minimal partition (SMP) problems using Dirichlet Laplacian eigenvalues considered by Helffer, Terracini and others. After a brief discussion of the history of the problem and existence and well-posedness results, we will show that, as the parameter $\alpha$ appearing in the boundary condition tends to zero, the minimal partitions converge (up to subsequences) to a minimal partition for the corresponding Cheeger partition problem, where the functional is based on a purely geometric quantity (the Cheeger constant) related to the isoperimetric ratio of the partition cells. This also has some interesting consequences for the eigenvalues of the Robin problem, for example as regards (a lack of) domain monotonicity. This talk will be based on an ongoing joint project with Pêdra Andrade, Nuno Carneiro, Matthias Hofmann, and Hugo Tavares, and (finished) work with João Ribeiro, in which we prove analogous results on metric graphs in place of domains.