27 research outputs found
On the parity of the number of nodal domains for an eigenfunction of the Laplacian on tori
In this note, we discuss a question posed by T. Hoffmann-Ostenhof concerning
the parity of the number of nodal domains for a non-constant eigenfunction of
the Laplacian on flat tori. We present two results. We first show that on the
torus , a non-constant eigenfunction has an
even number of nodal domains. We then consider the torus
, with
, and construct on it an eigenfunction with three
nodal domains.Comment: 5 pages, 2 figure
Courant-sharp eigenvalues of the three-dimensional square torus
In this paper, we determine, in the case of the Laplacian on the flat
three-dimensional torus , all the eigenvalues having
an eigenfunction which satisfies the Courant nodal domains theorem with
equality (Courant-sharp situation). Following the strategy of {\AA}. Pleijel
(1956), the proof is a combination of an explicit lower bound of the counting
function and a Faber-Krahn-type inequality for domains on the torus, deduced
as, in the work of P. B\'erard and D. Meyer (1982), from an isoperimetric
inequality. This inequality relies on the work of L. Hauswirth, J. Perez, P.
Romon, and A. Ros (2004) on the periodic isoperimetric problem.Comment: 9 pages, 1 tabl
Spectral minimal partitions for a family of tori
We study partitions of the rectangular two-dimensional flat torus of length 1
and width b into k domains, with b a parameter in (0, 1] and k an integer. We
look for partitions which minimize the energy, definedas the largest first
eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are
inparticular interested in the way these minimal partitions change when b is
varied. We present herean improvement, when k is odd, of the results on
transition values of b established by B. Helffer andT. Hoffmann-Ostenhof (2014)
and state a conjecture on those transition values. We establishan improved
upper bound of the minimal energy by explicitly constructing hexagonal tilings
of thetorus. These tilings are close to the partitions obtained from a
systematic numerical study based on an optimization algorithm adapted from B.
Bourdin, D. Bucur, and {\'E}. Oudet (2009). These numerical results also
support our conjecture concerning the transition values and give
betterestimates near those transition values
Majoration des valeurs propres Courant strictes de Neumann et Robin
30 pages. To appear in "Bulletin de la SociĂ©tĂ© MathĂ©matique de France".We consider the eigenvalues of the Laplacian on an open, bounded, connected set in R n with C 2 boundary, with a Neumann boundary condition or a Robin boundary condition. We obtain upper bounds for those eigenvalues that have a corresponding eigenfunction which achieves equality in Courant's Nodal Domain theorem. In the case where the set is also assumed to be convex, we obtain explicit upper bounds in terms of some of the geometric quantities of the set. MSC classification (2010): 35P15, 49R05, 35P05.Nous considĂ©rons les valeurs propres du laplacien sur un ouvert bornĂ© connexe de R n Ă bord C 2 , avec condition au bord de Neumann ou de Robin. Nous majorons celles qui ont une fonction propre dont le nombre de domaines nodaux atteint la borne de Courant (dites Courant strictes).Lorsque lâouvert est convexe, nous prĂ©sentons une majoration explicite en fonction de grandeurs gĂ©omĂ©triques
Concrete method for recovering the Euler characteristic of quantum graphs
Trace formulas play a central role in the study of spectral geometry and in
particular of quantum graphs. The basis of our work is the result by Kurasov
which links the Euler characteristic of metric graphs to the spectrum of
their standard Laplacian. These ideas were shown to be applicable even in an
experimental context where only a finite number of eigenvalues from a physical
realization of quantum graph can be measured.
In the present work we analyse sufficient hypotheses which guarantee the
successful recovery of . We also study how to improve the efficiency of
the method and in particular how to minimise the number of eigenvalues
required. Finally, we compare our findings with numerical
examples---surprisingly, just a few dozens of eigenvalues can be enough.Comment: 18 pages, 4 figures. Accepted by "Journal of Physics A: Mathematical
and Theoretical
Estimates for the lowest Neumann eigenvalues of parallelograms and domains of constant width
We prove sharp upper bounds for the first and second non-trivial eigenvalues
of the Neumann Laplacian in two classes of domains: parallelograms and domains
of constant width. This gives in particular a new proof of an isoperimetric
inequality for parallelograms recently obtained by A. Henrot, A. Lemenant and
I. Lucardesi.Comment: 17 pages, 1 figure. Proposition 4.5 adde
A theory of spectral partitions of metric graphs
We introduce an abstract framework for the study of clustering in metric
graphs: after suitably metrising the space of graph partitions, we restrict
Laplacians to the clusters thus arising and use their spectral gaps to define
several notions of partition energies; this is the graph counterpart of the
well-known theory of spectral minimal partitions on planar domains and includes
the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012),
815--838] as a special case. We focus on the existence of optimisers for a
large class of functionals defined on such partitions, but also study their
qualitative properties, including stability, regularity, and parameter
dependence. We also discuss in detail their interplay with the theory of nodal
partitions. Unlike in the case of domains, the one-dimensional setting of
metric graphs allows for explicit computation and analytic -- rather than
numerical -- results. Not only do we recover the main assertions in the theory
of spectral minimal partitions on domains, as studied in [Conti \textit{et al},
Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\
Henri Poincar\'e Anal.\ Non Lin\'eaire \textbf{26} (2009), 101--138], but we
can also generalise some of them and answer (the graph counterparts of) a few
open questions
Spectral minimal partitions of a sector
26 pagesInternational audienceIn this article, we are interested in determining the spectral minimal -partition for angular sectors. We first deal with the nodal cases for which we can determine explicitly the minimal partitions. Then, in the case where the minimal partitions are not given by eigenfunctions of the Dirichlet Laplacian, we analyze the possible topologies of the minimal partitions. We first exhibit symmetric minimal partitions by using mixed Dirichlet-Neumann Laplacian and then use a double covering approach to catch non symmetric candidates