142 research outputs found
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
Nodal and spectral minimal partitions -- The state of the art in 2015 --
In this article, we propose a state of the art concerning the nodal and
spectral minimal partitions. First we focus on the nodal partitions and give
some examples of Courant sharp cases. Then we are interested in minimal
spectral partitions. Using the link with the Courant sharp situation, we can
determine the minimal k-partitions for some particular domains. We also recall
some results about the topology of regular partitions and Aharonov-Bohm
approach. The last section deals with the asymptotic behavior of minimal
k-partition
Harmonic oscillators with Neumann condition on the half-line
International audienceThis paper is devoted to the computation of the minimum of the first eigenvalues for the Neumann realization of harmonic oscillators on the half-line. We propose an algorithm to determine this minimum and we estimate the accuracy of these computations. We also give numerical computations of constants appearing in superconductivity theory
Holomorphic extension of the de Gennes function
This note is devoted to prove that the de Gennes function has a holomorphic
extension on a strip containing the real axis
Magnetic WKB Constructions
This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB
expansions for the eigenfunctions were only established in presence of a
non-zero electric potential. Here we tackle the pure magnetic case. Thanks to
Feynman-Hellmann type formulas and coherent states decomposition, we develop
here a magnetic Born-Oppenheimer theory. Exploiting the multiple scales of the
problem, we are led to solve an effective eikonal equation in pure magnetic
cases and to obtain WKB expansions. We also investigate explicit examples for
which we can improve our general theorem: global WKB expansions, quasi-optimal
estimates of Agmon and upper bound of the tunelling effect (in symmetric
cases). We also apply our strategy to get more accurate descriptions of the
eigenvalues and eigenfunctions in a wide range of situations analyzed in the
last two decades
On the negative spectrum of the Robin Laplacian in corner domains
For a bounded corner domain , we consider the Robin Laplacian in
with large Robin parameter. Exploiting multiscale analysis and a
recursive procedure, we have a precise description of the mechanism giving the
ground state of the spectrum. It allows also the study of the bottom of the
essential spectrum on the associated tangent structures given by cones. Then we
obtain the asymptotic behavior of the principal eigenvalue for this singular
limit in any dimension, with remainder estimates. The same method works for the
Schr\"odinger operator in with a strong attractive
delta-interaction supported on . Applications to some Erhling's
type estimates and the analysis of the critical temperature of some
superconductors are also provided
Ground Energy of the Magnetic Laplacian in Polyhedral Bodies
The asymptotic behavior of the first eigenvalues of magnetic Laplacian
operators with large magnetic fields and Neumann realization in polyhedral
domains is characterized by a hierarchy of model problems. We investigate
properties of the model problems (continuity, semi-continuity, existence of
generalized eigenfunctions). We prove estimates for the remainders of our
asymptotic formula. Lower bounds are obtained with the help of a classical IMS
partition based on adequate coverings of the polyhedral domain, whereas upper
bounds are established by a novel construction of quasimodes, qualified as
sitting or sliding according to spectral properties of local model problems.Comment: 59 page
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