On nodal domains and spectral minimal partitions: a survey

On nodal domains and spectral minimalpartitions: a surveyB. 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) anda partition of Ω by k open sets ω , we can consider the quantityjmax λ(ω ) where λ(ω ) is the ground state energy of the Dirichletj j jrealization of the Laplacian in ω . If we denote byL (Ω) thej kinfimum over all the k-partitions of max λ(ω ), a minimalj jk-partition is then a partition which realizes the infimum.Although the analysis is rather standard when k = 2 (we find thenodal domains of a second eigenfunction), the analysis of higherk’s becomes non trivial and quite interesting.In this talk, we consider the two-dimensional case and discuss theproperties of minimal spectral partitions, illustrate the difficultiesby considering simple cases like the disc, the rectangle or thesphere (k = 3) and will also exhibit the possible role of thehexagone in the asymptotic behavior as k → +∞ ofL (Ω).kWe also compare different notions of minimal partitions.This work has started in collaboration with T. Hoffmann-Ostenhofand 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 inbounded domains. We would like to analyze the relations ...