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.

Introduction

We are interested in the nodal domains of the eigenfunctions of the two-dimensional Dirichlet Laplacians.
We try to determine the relations between these domains and the partitions of the domains.

Definitions

Let Ω be a bounded and regular domain in R². The eigenvalues of the Dirichlet realization of the Laplacian -Δ in Ω are denoted by
λ₁ < λ₂ ≤ λ₃ ≤ ... ≤ λ_n ... We choose the associated eigenfunctions u_n to form an orthonormal basis for L²(Ω).
Let k ≥ 1 be an integer. A partition of Ω is a family D = { D_i }^k_i=1 of mutually disjoint sets such that
&cup^k_i=1 D_i ⊂ Ω
The partition is called open, resp. connected, regular, if the D_i are open, resp. connected, regular (i. e. piecewise C¹ and with the interior cone condition) sets of Ω. The partition is called strong if

Int (

∪_i D_i

) \ ∂ Ω = Ω.

The set of open connected partitions is denoted D_k.

For D ∈ D_k, we define
Λ( D ) = max { λ(D_i), i=1,...,k },
where λ(D_i) denotes the first eigenvalue of the Dirichlet realization of the Laplacian in D_i.
For any integer k ≥ 1, we define
L_k = inf { Λ( D ), D &isin D_k }.
A partition D &isin D_k such that Λ( D ) = L_k is called k-minimal partition.
Let u ∈ C⁰₀(Ω). The nodal domains of u (whose number is denoted μ(u)) are the components of Ω \ N(u) where

N(u) =

{ x ∈ Ω | u(x) = 0 }

Results about minimal partitions

2-partitions (cf. [B. Helffer, T. Hoffmann-Ostenhof], [M. Conti, S. Terracini, G. Verzini])

L₂ is the second eigenvalue λ₂ and the minimal 2-partition is the nodal partition associated to the second eigenvector (so μ(u₂)=2).
Remark: we naturally wonder wether any minimal partition is a nodal induced by an eigenfunction.

Existence (cf. [M. Conti, S. Terracini, G. Verzini])

For any integer k, there exists a k-minimal regular strong parition.
Conversely, we would like to know if any k-minimal partition is regular and strong.

Uniqueness for subpartitions

Let D be a k-minimal partition relative to L_k. Let Ω' ⊂ Ω be connected and D' ⊂ D be any subpartition of D into k' elements (1 ≤ k' < k) such that

Ω'

= ∪ {

D_i

, D_i ∈ D' }.

Then L_k (Ω) = L_k' (Ω') and this last equality is uniquely achieved.

Bipartite partitions

Notations

We say that D_i, D_j are neighbors ( D_i ∼ D_j ) if Int (

D_i ∪ D_j

) \ ∂ Ω is connected.

We associate to each D a graph G( D ) by associating to each D_i a vertex and to each pair D_i ∼ D_j an edge. This is an undirected graph without multiple edges or loops.
The graph is said bipartite if it can be colored by two colors.

Theorem

If the graph of the minimal partition is bipartite, this is a partition associated to the nodal set of an eigenfunction corresponding to L_k.

Courant's nodal theorem

Let k ≥ 1, λ_k be the k-th eigenvalue and u any real eigenfunction of -Δ on Ω (so that -Δ u = λ_k u). Then the number of nodal domains μ(u) of u satisfies &mu(u) ≤ k.

Courant-sharp function

Let u be an eigenfunction of the Dirichlet realization of the Schrödinger operator -Δ+V in Ω associated to λ_k (i. e. (-Δ+V)u = λ_k ). If μ(u) = k, we say that u is Courant-sharp.
For any integer k ≥ 1, we denote by L_k the smallest eigenvalue whose eigenspace contains an eigenfunction with k nodal domains. In general (to be precised...), λ_k ≤ L_k ≤ L_k.
We would like to determine the equality cases.

Theorem

We assume that Ω is regular. If L_k = L_k, then λ_k = L_k = L_k. Furthermore, there exists an eigenfunction u_k in the eigenspace associated to λ_k such that μ(u_k) = k (i.e. u_k is Courant-sharp).

Simulations

Eigenmodes on triangles and hexagones.

Second eigenmode on hemi-domains with moving Dirichlet condition.

Transition from Courant sharp to a non nodal minimal partition on rectangles.

Iterative algorithm to determine some candidate for the minimal 3-partition.

Eigenmodes on the double covering : square and regular polygons, square with moving interior point.

References

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Dernière mise à jour le 09 juin 2008