528 research outputs found
A necessary condition for lower semicontinuity of line energies
We are interested in some energy functionals concentrated on the
discontinuity lines of divergence-free 2D vector fields valued in the circle
. This kind of energy has been introduced first by P. Aviles and
Y. Giga. They show in particular that, with the cubic cost function ,
this energy is lower semicontinuous. In this paper, we construct a
counter-example which excludes the lower semicontinuity of line energies for
cost functions of the form with . We also show that, in this case,
the viscosity solution corresponding to a certain convex domain is not a
minimizer.Comment: 13 page
Approximations elliptiques d'Ă©nergies singuliĂšres sous contrainte de divergence
This thesis is devoted to the study of phase-field type variational models with divergence constraint. These models typically involve an energy depending on a parameter which represents a negligible physical quantity or is linked to some numerical approximation method for instance. A central question concerns the asymptotic behavior of these energies and of their global or local minimizers when this parameter goes to 0. We present different strategies which allow to take the divergence constraint into account. They will be illustrated in two models. The first one is a phase-field type approximation, involving a divergence constraint, of the Eulerian model for branched transportation. We illustrate how uniform estimates on the energy, depending on the constraint on the divergence, allow to establish a Gamma-convergence result. The second model, related to micromagnetics, concerns Aviles-Giga type energies for divergence-free vector fields. We use the entropy method in order to characterize global minimizers. In some situations, we will prove a De Giorgi type conjecture concerning the one-dimensional symmetry of global minimizers under boundary conditions.Cette theÌse est consacreÌe aÌ lâeÌtude de certains probleÌmes variationnels de type transition de phase vectorielle ou "phase-field" qui font intervenir une contrainte de divergence. Ces modeÌles sont geÌneÌralement baseÌs sur une eÌnergie deÌpendant dâun parameÌtre qui peut repreÌsenter une grandeur physique neÌgligeable ou qui est lieÌe aÌ une meÌthode dâapproximation numeÌrique par exemple. Une question centrale concerne alors le comportement asymptotique de ces eÌnergies et des minimiseurs globaux ou locaux lorsque ce parameÌtre tend vers 0. Cette theÌse preÌsente diffeÌrentes strateÌgies prenant en compte la contrainte de divergence. Elles seront illustreÌes aÌ travers lâeÌtude de deux modeÌles. Le premier est une approximation du modeÌle EuleÌrien pour le transport brancheÌ par un modeÌle de type phase-field avec divergence prescrite. Nous montrons comment une estimation uniforme de lâeÌnergie, en fonction de la contrainte sur la divergence, permet dâeÌtablir un reÌsultat de Gamma-convergence. Le second modeÌle, en lien avec la theÌorie du micromagneÌtisme, concerne des eÌnergies de type Aviles-Giga dans un cadre vectoriel avec contrainte de divergence. Nous illustrerons dans quelle mesure la meÌthode dâentropie permet de caracteÌriser les minimiseurs globaux. Dans certaines situations nous montrerons une conjecture de type De Giorgi concernant la symeÌtrie 1D des minimiseurs globaux de lâeÌnergie sous une contrainte au bord
Ginzburg-Landau relaxation for harmonic maps on planar domains into a general compact vacuum manifold
We study the asymptotic behaviour, as a small parameter tends
to zero, of minimisers of a Ginzburg-Landau type energy with a nonlinear
penalisation potential vanishing on a compact submanifold and
with a given -valued Dirichlet boundary data. We show that
minimisers converge up to a subsequence to a singular -valued
harmonic map, which is smooth outside a finite number of points around which
the energy concentrates and whose singularities' location minimises a
renormalised energy, generalising known results by Bethuel, Brezis and H\'elein
for the circle . We also obtain -convergence results and
uniform Marcinkiewicz weak or Lorentz estimates on the derivatives.
We prove that solutions to the corresponding Euler-Lagrange equation converge
uniformly to the constraint and converge to harmonic maps away from
singularities.Comment: 41 pages, minor revisio
Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains
We define renormalised energies for maps that describe the first-order
asymptotics of harmonic maps outside of singularities arising due to
obstructions generated by the boundary data and the mutliple connectedness of
the target manifold. The constructions generalise the definition by Bethuel,
Brezis and H\'elein for the circle (Ginzburg-Landau vortices, 1994). In
general, the singularities are geometrical objects and the dependence on
homotopic singularities can be studied through a new notion of synharmony. The
renormalised energies are showed to be coercive and Lipschitz-continuous. The
renormalised energies are associated to minimising renormalisable singular
harmonic maps and minimising configurations of points can be characterised by
the flux of the stress-energy tensor at the singularities. We compute the
singular energy and the renormalised energy in several particular cases.Comment: 38 pages, bibliography update
A necessary condition for lower semicontinuity of line energies
Abstract We are interested in some energy functionals concentrated on the discontinuity lines of divergence-free 2D vector fields valued in the circle S 1 . This kind of energy has been introduced first by P. Aviles and Y. Giga i
Semaine d'Etude MathĂ©matiques et Entreprises 6 : Analyse et filtrage temps-freÌquence de "bursts" ultrasonores : identification, classification, seÌparation
Ce rapport est une preÌsentation de nos reÌsultats et de nos reÌflexions aÌ propos du probleÌme proposeÌ par IFP EÌnergies Nouvelles pendant la sixieÌme eÌdition de la Semaine d'EÌtude Maths-Entreprises. Nous disposions d'enregistrements de bursts ultrasonores issus d'un probleÌme physique de corrosion d'eÌprouvettes en meÌtal. Le but eÌtait de donner une classification des signaux acoustiques visant aÌ identifier les diffeÌrentes typologies de corrosion. En Section 1 on trouve une preÌsentation plus deÌtailleÌe de la probleÌmatique enqueÌteÌe. Tous les approches consideÌreÌes sont deÌveloppeÌes dans la Section 3, alors que dans la Section 2 on a passeÌ en revue les outils matheÌmatiques neÌcessaires
Measurement of the top quark forward-backward production asymmetry and the anomalous chromoelectric and chromomagnetic moments in pp collisions at âs = 13 TeV
Abstract The parton-level top quark (t) forward-backward asymmetry and the anomalous chromoelectric (dÌ t) and chromomagnetic (ÎŒÌ t) moments have been measured using LHC pp collisions at a center-of-mass energy of 13 TeV, collected in the CMS detector in a data sample corresponding to an integrated luminosity of 35.9 fbâ1. The linearized variable AFB(1) is used to approximate the asymmetry. Candidate t t ÂŻ events decaying to a muon or electron and jets in final states with low and high Lorentz boosts are selected and reconstructed using a fit of the kinematic distributions of the decay products to those expected for t t ÂŻ final states. The values found for the parameters are AFB(1)=0.048â0.087+0.095(stat)â0.029+0.020(syst),ÎŒÌt=â0.024â0.009+0.013(stat)â0.011+0.016(syst), and a limit is placed on the magnitude of | dÌ t| < 0.03 at 95% confidence level. [Figure not available: see fulltext.
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