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On nodal domains and spectral minimal partitions: a survey

57 pages
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 ...
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On nodal domains and spectral minimal partitions: a survey
B. Helffer (Univ Paris-Sud and CNRS)
(AfterV.Bonnaillie-No¨el,B.Heler,T.Homann-Ostenhof, S. Terracini, G. Vial)
Franco-Egyptian meeting : 2 of May 2010
Given a bounded open setΩinRn(or a Riemannian manifold) and a partition ofΩbykopen setsωj, we can consider the quantity maxjλ(ωj)whereλ(ωj)is the ground state energy of the Dirichlet realization of the Laplacian inωj. If we denote byLk(Ω)the infimum over all thek-partitions ofmaxjλ(ωj), 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+ofLk(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 setsDiwhich are minimal in the sense that the maximum over theDi’s of the ground state energy of the Dirichlet realization of the Laplacian inDiis minimal.
LetΩ Letbe a regular bounded domain.H(Ω)be the Laplacian ΔonΩR2with Dirichlet boundary condition (uΩ= 0). In the case of a Riemannian manifold we will consider the Laplace Beltrami operator. We denote byλj(Ω)the increasing sequence of its eigenvalues and byujsome associated orthonormal basis of eigenfunctions. The groundstateu1can be chosen to be strictly positive inΩ, but the other eigenfunctionsukmust have zerosets. We define for anyuC00(Ω)
N(u) =
{xΩu(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]
Letk1,λkbe thek-th eigenvalue andE(λk)the eigenspace of H(Ω)associated toλk. Then,uE(λk)\ {0} µ(u)k
Theorem [Pleijel]
There existsk0such that ifkk0, then
µ(u)<kuE(λk)\ {0}
Proposition a
For any eigenvalueλofH(Ω)corresponding to an eigenfunctionu withknodal domains we have
πj2 λk|Ω|
(2)
where|Ω|denotes the area ofΩandjis the smallest positive zero of the Bessel functionJ0.
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)|πωj2|
(3)
Ifuis an eigenfunction ofHattached to the eigenvalueλwithk nodal sets then we have for any of these nodal domainsDi:
That we rewrite in the form
Summing overi, we get
λ|πjDi2|
|Di| ≥πj2 λ 
|Ω| ≥kπj2λ
(4)
(5)
The Weyl theory says that
asn+.
λ4πn n|Ω|
Ifnis large (6) and (2), having in mind the value of uncan not havennodal domains !
(6)
j2404. So
Partitions
We first introduce the notion of partition.
Definition 1
Let1kN will call. Wepartition(ork-partition for indicating the cardinal of the partition) ofΩa familyD={Di}ki=1of mutually disjoint sets such that
ki=1DiΩ
(7)
We call itopenif theDiare open sets ofΩ,connectedif theDi are connected.
We denote byOkthe 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 integerk1, and forDinOk, we introduce
Then we define
Λ(D) = maxλ(Di)i
Lk= inf Λ(D)D∈Ok and callD ∈Okminimal ifLk= Λ(D).
(8)
(9)