Abstract
Given a bounded open set Ω in \({\mathbb{R}^n}\) (or a Riemannian manifold) and a partition of Ω by k open sets D j , we can consider the quantity max j λ(D j ) where λ(D j ) is the groundstate 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 λ(D j ), a minimal (spectral) 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 survey, we mainly consider the two-dimensional case and discuss the properties of minimal spectral partitions.We illustrate the difficulties to determine them explicitly by considering simple cases like the disk, the rectangle or the sphere (k = 3) and will also exhibit the possible role of the hexagon in the asymptotic behavior as \({k\,\to\, +\infty\, {\rm of}\, \mathfrak{L}_k(\Omega)}\) .
We also discuss the role of some Aharonov-Bohm Schrödinger operator for producing candidates for minimal partitions. Finally we compare different notions of minimal partitions and propose a few open problems.
This work has started in collaboration with T. Hoffmann-Ostenhof and has been continued in the last years with the coauthors V. Bonnaillie-Noël, T. Hoffmann- Ostenhof, S. Terracini and G. Vial.
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Lecture held in the Seminario Matematico e Fisico on October 26, 2009.
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Helffer, B. On Spectral Minimal Partitions: A Survey. Milan J. Math. 78, 575–590 (2010). https://doi.org/10.1007/s00032-010-0129-0
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DOI: https://doi.org/10.1007/s00032-010-0129-0