Seminar

# Minimal partitions and Aharonov-Bohm hamiltonians

#### Bernard Helffer - Université de Nantes

Given a bounded open set $\Omega$ in $\mathbb R^n$ (or in a Riemannian manifold) and a partition $\mathcal D$ of by $k$ open sets $\clr D_j$, we can consider the quantity $\Lambda (\mathcal D):= max_j\lambda (D_j)$ where $\lambda (D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$. If we denote by $\mathfrak L_k(\Omega)$ the infimum over all the $k$-partitions of $\Lambda(\mathcal D)$ a minimal $k$-partition is then a partition which realizes the infimum. Although the analysis is rather standard when $k = 2$ (we find the nodal domains of a second eigenfunction), the analysis of higher k's becomes non trivial and quite interesting. In this talk, we consider the two-dimensional case and discuss the properties of minimal spectral partitions, illustrate the difficulties by considering simple cases like the rectangle, the torus, the disk (...) and then give a "magnetic" characterization of these minimal partitions using Aharonov-Bohm operators. This work has started in collaboration with T. Hoffmann-Ostenhof (with a preliminary work with M. and T. Hoffmann-Ostenhof and M. Owen) and has been continued with him and other coauthors : V. Bonnaillie-No\"el, S. Terracini, G. Vial, and C. Lena
Organizers
Rafael D. Benguria