On nodal domains and spectral minimal partitions: a survey

Description

On nodal domains and spectral minimal
partitions: a survey
B. Helffer (Univ Paris-Sud and CNRS)
(After V. Bonnaillie-No¨el, B. Helffer, T. Hoffmann-Ostenhof,
S. Terracini, G. Vial)
Franco-Egyptian meeting : 2 of May 2010nGiven a bounded open set Ω inR (or a Riemannian manifold) and
a partition of Ω by k open sets ω , we can consider the quantityj
max λ(ω ) where λ(ω ) is the ground state energy of the Dirichletj j j
realization of the Laplacian in ω . If we denote byL (Ω) thej k
infimum over all the k-partitions of max λ(ω ), a minimalj j
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 talk, we consider the two-dimensional case and discuss the
properties of minimal spectral partitions, illustrate the difficulties
by considering simple cases like the disc, the rectangle or the
sphere (k = 3) and will also exhibit the possible role of the
hexagone in the asymptotic behavior as k → +∞ ofL (Ω).k
We also compare different notions of minimal partitions.
This work has started in collaboration with T. Hoffmann-Ostenhof
and has been continued (published, to appear or in preparation)
with the coauthors mentioned above : V. Bonnaillie-No¨el,
T. Hoffmann-Ostenhof, S. Terracini, G. Vial.We consider mainly two-dimensional Laplacians operators in
bounded domains. We would like to analyze the relations ...

Sujets

• Partition héraldique
• Nodale
• MathML
• Preuve de concept
• Candidate for Goddess
• Méthode des éléments finis de frontière

Informations

• Publié par: Fyom
• Nombre de lectures: 87
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

On nodal domains and spectral minimal partitions: a survey

B. Helffer (Univ Paris-Sud and CNRS)

(AfterV.Bonnaillie-No¨el,B.Helffer,T.Hoffmann-Ostenhof, S. Terracini, G. Vial)

Franco-Egyptian meeting : 2 of May 2010

Given a bounded open setΩinR n (or a Riemannian manifold) and a partition ofΩbykopen setsω j , we can consider the quantity max j λ(ω j )whereλ(ω j )is the ground state energy of the Dirichlet realization of the Laplacian inω j . If we denote byL k (Ω)the infimum over all thek-partitions ofmax j λ(ω j ), a minimal k-partition is then a partition which realizes the infimum. Although the analysis is rather standard whenk= 2(we find the nodal domains of a second eigenfunction), the analysis of higher k’s becomes non trivial and quite interesting.

In this talk, we consider the two-dimensional case and discuss the properties of minimal spectral partitions, illustrate the difficulties by considering simple cases like the disc, the rectangle or the sphere (k= 3also exhibit the possible role of the) and will hexagone in the asymptotic behavior ask→+∞ofL k (Ω). We also compare different notions of minimal partitions. This work has started in collaboration with T. Hoffmann-Ostenhof and has been continued (published, to appear or in preparation) withthecoauthorsmentionedabove:V.Bonnaillie-Noe¨l, T. Hoffmann-Ostenhof, S. Terracini, G. Vial.

We consider mainly two-dimensional Laplacians operators in bounded domains. We would like to analyze the relations between the nodal domains of the eigenfunctions of the Dirichlet Laplacians and the partitions bykopen setsD i which are minimal in the sense that the maximum over theD i ’s of the ground state energy of the Dirichlet realization of the Laplacian inD i is minimal.

LetΩ Letbe a regular bounded domain.H(Ω)be the Laplacian −ΔonΩ⊂R 2 with Dirichlet boundary condition (u ∂Ω = 0). In the case of a Riemannian manifold we will consider the Laplace Beltrami operator. We denote byλ j (Ω)the increasing sequence of its eigenvalues and byu j some associated orthonormal basis of eigenfunctions. The groundstateu 1 can be chosen to be strictly positive inΩ, but the other eigenfunctionsu k must have zerosets. We define for anyu∈C 00 (Ω)

N(u) =

{x∈Ω  u(x) = 0}

(1)

and call the components ofΩ\N(u)the nodal domains ofu. The number of nodal domains ofuis calledµ(u). Thesek=µ(u) nodal domains define a partition ofΩ.

The Courant nodal theorem says :

Theorem [Courant]

Letk≥1,λ k be thek-th eigenvalue andE(λ k )the eigenspace of H(Ω)associated toλ k . Then,∀u∈E(λ k )\ {0} µ(u)≤k

Theorem [Pleijel]

There existsk 0 such that ifk≥k 0 , then

µ(u)<k∀u∈E(λ k )\ {0}

Proposition a

For any eigenvalueλofH(Ω)corresponding to an eigenfunctionu withknodal domains we have

πj 2 λ≥k |Ω| 

(2)

where|Ω|denotes the area ofΩandjis the smallest positive zero of the Bessel functionJ 0 .

The proof is actually a side result of the proof by Pleijel’s of his theorem [Pl]. The main point is the Faber-Krahn Inequality :

λ(ω)≥ | π ω j 2 | 

(3)

Ifuis an eigenfunction ofHattached to the eigenvalueλwithk nodal sets then we have for any of these nodal domainsD i :

That we rewrite in the form

Summing overi, we get

λ≥ | πj D i 2 |

|D i | ≥ πj 2 λ 

|Ω| ≥k πj 2  λ

(4)

(5)

The Weyl theory says that

asn→+∞.

λ∼ 4πn n |Ω| 

Ifnis large (6) and (2), having in mind the value of u n can not havennodal domains !

(6)

j∼2404. So

Partitions

We first introduce the notion of partition.

Definition 1

Let1≤k∈N will call. Wepartition(ork-partition for indicating the cardinal of the partition) ofΩa familyD={D i } k i=1 of mutually disjoint sets such that

∪ k i=1 D i ⊂Ω

(7)

We call itopenif theD i are open sets ofΩ,connectedif theD i are connected.

We denote byO k the set of open connected partitions. Sometimes (at least for the proof) we have to relax this definition by considering quasi-open or measurable sets for the partitions.

We now introduce the notion of spectral minimal partition sequence.

Definition 2

For any integerk≥1, and forDinO k , we introduce

Then we define

Λ(D) = maxλ(D i ) i

L k = inf Λ(D) D∈O k and callD ∈O k minimal ifL k = Λ(D).

(8)

(9)