Nodal domains and spectral minimal partitions

We recall here some theoretical results of B. Helffer, T. Hoffmann-Ostenhof and S. Terracini.

We propose numerical computations to illustrate some conjectures.

We try to determine the relations between these domains and the partitions of the domains.

Let k ≥ 1 be an integer. A partition of Ω is a family D = { D

The partition is called open, resp. connected, regular, if the D

Int ( | ∪_{i} D_{i} |
) \ ∂ Ω = Ω. |

The set of open connected partitions is denoted D

For D ∈ D

where λ(D

For any integer k ≥ 1, we define

A partition D &isin D

Let u ∈ C

N(u) = | { x ∈ Ω | u(x) = 0 } | . |

Conversely, we would like to know if any k-minimal partition is regular and strong.

Ω' | = ∪ { | D_{i} |
, D_{i} ∈ D' }. |

We say that D_{i}, D_{j} are neighbors ( D_{i} ∼ D_{j} ) if Int ( |
D_{i} ∪ D_{j} |
) \ ∂ Ω is connected. |

The graph is said bipartite if it can be colored by two colors.

For any integer k ≥ 1, we denote by L

We would like to determine the equality cases.

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