# 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 ≤ ... ≤ λn ...
We choose the associated eigenfunctions un to form an orthonormal basis for L2(Ω).
Let k ≥ 1 be an integer. A partition of Ω is a family D = { Di }ki=1 of mutually disjoint sets such that
&cupki=1 Di ⊂ Ω

The partition is called open, resp. connected, regular, if the Di 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 Di ) \ ∂ Ω = Ω.

The set of open connected partitions is denoted Dk.

For D Dk, we define
Λ( D ) = max { λ(Di), i=1,...,k },

where λ(Di) denotes the first eigenvalue of the Dirichlet realization of the Laplacian in Di.
For any integer k ≥ 1, we define
Lk = inf { Λ( D ), D &isin Dk }.

A partition D &isin Dk such that Λ( D ) = Lk 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 } .

## 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 Lk. Let Ω' ⊂ Ω be connected and D' D be any subpartition of D into k' elements (1 ≤ k' < k) such that
 Ω' = ∪ { Di , Di ∈ D' }.
Then Lk (Ω) = Lk' (Ω') and this last equality is uniquely achieved.

# Bipartite partitions

## Notations

 We say that Di, Dj are neighbors ( Di ∼ Dj ) if Int ( Di ∪ Dj ) \ ∂ Ω is connected.
We associate to each D a graph G( D ) by associating to each Di a vertex and to each pair Di ∼ Dj 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 Lk.

## Courant's nodal theorem

Let k ≥ 1, λk be the k-th eigenvalue and u any real eigenfunction of -Δ on Ω (so that -Δ u = λk 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 λk (i. e. (-Δ+V)u = λk ). If μ(u) = k, we say that u is Courant-sharp.
For any integer k ≥ 1, we denote by Lk the smallest eigenvalue whose eigenspace contains an eigenfunction with k nodal domains. In general (to be precised...),
λk Lk ≤ Lk.

We would like to determine the equality cases.

## Theorem

We assume that Ω is regular. If Lk = Lk, then
λk = Lk = Lk.
Furthermore, there exists an eigenfunction uk in the eigenspace associated to λk such that μ(uk) = k (i.e. uk 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