Open AccessIntroduction to some conjectures for spectral minimal partitions
Author(s) -
Bernard Helffer
Publication year - 2011
Publication title -
journal of the egyptian mathematical society
Language(s) - English
Resource type - Journals
eISSN - 2090-9128
pISSN - 1110-256X
DOI - 10.1016/j.joems.2011.09.003
Subject(s) - mathematics , infimum and supremum , partition (number theory) , rectangle , bounded function , combinatorics , eigenfunction , embedding , laplace operator , open set , pure mathematics , mathematical analysis , geometry , eigenvalues and eigenvectors , physics , quantum mechanics , artificial intelligence , computer science
Given a bounded open set Ω in Rn (or in a Riemannian manifold) and a partition of Ω by k open sets Dj, we consider the quantity maxjλ(Dj) where λ(Dj) is the ground state energy of the Dirichlet realization of the Laplacian in Dj. If we denote by Lk(Ω) the infimum over all the k-partitions of maxjλ(Dj), a minimal k-partition is then a partition which realizes the infimum. When k=2, we find the two nodal domains of a second eigenfunction, but the analysis of higher k’s is non trivial and quite interesting. In this paper, which is complementary of the survey [20], we consider the two-dimensional case and present the properties of minimal spectral partitions, illustrate the difficulties by considering simple cases like the disk, the rectangle or the sphere (k=3). We will present also the main conjectures in this rather new subject