On Nodal domains and spectral minimal partitions: new results and open problems

Given a bounded open set $\Omega$
in $\mathbb{R}^2$ or in a Riemannain manifold
and a partition of $\Omega$
by $k$ open sets $\omega_j$ , we can consider the quantity
$\max_j \lambda(\omega_j)$ where $\lambda(\omega_j)$ is the ground state energy of
the Dirichlet realization of the Laplacian in $\omega_j$. If we denote by
$\mathcal{L}_k(\Omega
)$ the infimum over all the $k$-partitions of
$\max_j \lambda(\omega_j)$, 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 $k$ becomes non trivial and quite interesting.
In this talk, we would like to discuss the properties of minimal spectral partitions,
illustrate the difficulties by considering simple cases like the disc, the rectangle
or the sphere ($k = 3$) and also exhibit the possible role of the hexagone in the
asymptotic behavior as $k \to \infty$ of
$\mathcal{L}_k(\Omega
)$. We will also explain the link of these questions with spectral
properties of some Aharonov-Bohm hamiltonians.
We will finally discuss other definitions of minimal partitions.
This work has started in collaboration with T. Hoffmann-Ostenhof and has been continued
(published or to appear) with the coauthors V. Bonnaillie-Noel, T. Hoffmann-Ostenhof, S. Terracini
and G. Vial.