Breadcrumb
On nodal domains and spectral minimal partitions: A survey
Mon 07 January 2013, 16:15
Bernard Helffer
Universite Paris Sud
Organiser: Michiel van den Berg
ABSTRACT
Given a bounded open set $\Omega$ in $\mathbb{R^n}$ (or in a Riemannian manifold) and a partition of $\Omega$ by $k$ open sets $D_j$, we can consider the quantity $\max_j \lambda(D_j)$ (which is called the energy of the partition) 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 $\max_j \lambda(D_j)$, 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 $k$'s becomes non trivial and quite interesting. In this talk, we consider the two-dimensional case and present the state of the art for open sets in $\mathbb R^2$ or for other examples like the two dimensional torus.