On nodal domains and spectral minimal partitions: a survey

On nodal domains and spectral minimal
partitions: a survey
B. Helffer (Univ Paris-Sud and CNRS)
(After V. Bonnaillie-No¨el, B. Helffer, T. Hoffmann-Ostenhof,
S. Terracini, G. Vial)
Franco-Egyptian meeting : 2 of May 2010nGiven a bounded open set Ω inR (or a Riemannian manifold) and
a partition of Ω by k open sets ω , we can consider the quantityj
max λ(ω ) where λ(ω ) is the ground state energy of the Dirichletj j j
realization of the Laplacian in ω . If we denote byL (Ω) thej k
infimum over all the k-partitions of max λ(ω ), a minimalj j
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 disc, the rectangle or the
sphere (k = 3) and will also exhibit the possible role of the
hexagone in the asymptotic behavior as k → +∞ ofL (Ω).k
We also compare different notions of minimal partitions.
This work has started in collaboration with T. Hoffmann-Ostenhof
and has been continued (published, to appear or in preparation)
with the coauthors mentioned above : V. Bonnaillie-No¨el,
T. Hoffmann-Ostenhof, S. Terracini, G. Vial.We consider mainly two-dimensional Laplacians operators in
bounded domains. We would like to analyze the relations ...