Séminaire d’analyse non linéaire
Organisateurs : Augusto C. Ponce, Jean Van Schaftingen & Michel Willem
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local b328, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Giacomo Canevari, (Université Pierre-et-Marie-Curie, Paris)
La formule de l'indice de Morse pour des champs de vecteurs VMO
Dans cet exposé, nous nous intéressons à la question suivante : étant donnée une variété compacte à bord \(N\subset\mathbb{R}^d\), quels sont les données au bord \(g\colon\partial N \to\mathbb{S}^{d-1}\) prolongeables à des champs de vecteurs unitaires sur \(N\) ? Dans le cadre continu, la formule de Morse donne une réponse possible à cette question en faisant intervenir un invariant topologique : l'indice d'un champ de vecteurs. Inspirés par les travaux de Brezis et Nirenberg, nous montrons comment ces notions s'étendent à des champs de régularité plus faible, appartenant à l'espace VMO (Vanishing Mean Oscillation). Enfin, nous donnons une application de ces résultats à des modèles de cristaux liquides nématiques étalés sur une surface.
Ce travail a été réalisé en collaboration avec A. Segatti et M. Veneroni.
local b328, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Alexandra Convent (Université catholique de Louvain)
Dérivées faibles intrinsèques d'applications entre variétés et espaces de Sobolev
Nous commençons par définir la notion de dérivée faible pour des applications entre variétés. Cette notion permet une définition intrinsèque des espaces de Sobolev entre variétés. Cette nouvelle définition est équivalente à la définition classique par plongement dans un espace euclidien et à celle des espaces de Sobolev à valeurs dans un espace métrique. Finalement, à nouveau grâce à cette notion de dérivées faibles, l'espace de Sobolev est muni de plusieurs distances intrinsèques qui induisent la même topologie et pour lesquelles l'espace est complet.
Ceci est un travail en collaboration avec Jean Van Schaftingen.
local b328, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Justin Dekeyser (UCL)
Autour de la convergence de polarisations aléatoires successives
Les réarrangements de fonctions permettent de résoudre des problèmes aux variations et d'établir nombre d'inégalités remarquables, telles que l'inégalité de Riesz ou l'inégalité de Pólya-Szegő. Une stratégie récurrente est l'approximation de fonctions totalement symétriques (symétrisées de Schwarz) via l'itération des polarisations. Ces dernières années, un intérêt particulier s'est porté sur l'approximation de la symétrisation de Schwarz par une suite de polarisations successives choisies de manière aléatoire. Dans cette présentation, nous montrons plusieurs résultats nouveaux concernant la convergence de polarisations aléatoires successives qui sont des processus de Markov homogènes dans le temps. Les résultats obtenus permettent de mieux comprendre le processus d'approximation universelle, tant sur sa stabilité que sur sa structure apparemment récurrente. Ils ouvrent également la voie vers d'autres perspectives encore inexplorées.
local b328, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Pierre Bousquet (Toulouse)
Sur quelques problèmes dégénérés en calcul des variations
Nous présentons quelques résultats de régularité pour deux problèmes en calcul des variations. Dans le premier problème, on établit un résultat de continuité lipschitzienne globale pour les minimiseurs quand le lagrangien est dégénéré (autrement dit, de hessienne non définie positive) seulement à l'intérieur d'une boule. Dans le second problème, la dégénérescence est localisée le long d'un nombre fini de droites et on prouve un résultat de continuité lipschitzienne locale.
Il s'agit d'un travail en collaboration avec Lorenzo Brasco et Vesa Julin.
local b328, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Nicolas Raymond (Université de Rennes)
Optimal magnetic Sobolev constants in the semiclassical limit
This talk is devoted to the semiclassical analysis of the best constants in the magnetic Sobolev embeddings in the case of a bounded domain of the plane carrying Dirichlet conditions. We provide quantitative estimates of these constants (with an explicit dependence on the semiclassical parameter) and analyze the exponential localization of the corresponding minimizers near the magnetic wells.
This is joint work with S. Fournais.
CYCL 08, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Gilles Evequoz (Frankfurt)
Existence and asymptotic properties of real-valued solutions of the nonlinear Helmholtz equation
CYCL 08, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Jean Van Schaftingen (UCL)
Nodal solutions for the Choquard equation
CYCL 08, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Laurent Moonens (Orsay)
Singularités éliminables des champs de vecteurs à divergence nulle dans des espaces de Lebesgue à poids
CYCL 08, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Juraj Foldes (ULB)
Long term dynamics of Euler equation and maximal entropy solutions
local CYCL02, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Luca Battaglia (Université catholique de Louvain)
The singular Liouville equation and Toda system on compact surfaces
I will talk about the singular Liouville equation on compact surfaces, that is a second-order elliptic PDE with an exponential nonlinearity. Then I will discuss the singular Toda system, which is a system of two equations with similar features. I will present some existence results obtained through variational methods.
Vitaly Moroz (Swansea University)
Density functional theory of charge screening in graphene
Thomas-Fermi-Dirac-von Weizsäcker type to describe the response of a single layer of graphene to a point charge or a collection of charges some distance away from the layer. We formulate a variational framework in which the proposed energy functional admits minimizers. The associated Euler-Lagrange equation for the charge density is obtained, and uniqueness, regularity and decay of the minimizers are proved under general conditions. In addition, a bifurcation from zero at a finite threshold value of the external charge is proved.
This is a joint work with Cyrill Muratov (NJIT) and Jianfeng Lu (Duke University).
CYCL09B
Antoine Gloria (ULB)
A large scale regularity for elliptic systems with random coefficients
CYCL09B
Wolfgang Reichel (KIT)
Electrostatic characterization of balls
CYCL09B
Stefan Krömer (Köln)
Heterogeneous thin films: local an nonlocal effects
CYCL09B
Alessio Pomponio (Pol. Bari)
Some results on the Chern-Simons-Schrödinger equation
CYCL 08
Armin Schikorra (Universität Freiburg)
Topology of the Heisenberg group and \(H=W\)-questions I
Approximation of Sobolev maps into Riemannian manifolds by smooth functions depends on the topology of the manifold. We report on recent progress of density questions for Sobolev maps into the Heisenberg group and related topological problems. The Heisenberg groups are examples of sub-Riemannian manifolds homeomorphic, but not diffeomorphic to the Euclidean space. Their metric is derived from curves which are only allowed to move in so-called horizontal directions.
CYCL 08
Armin Schikorra (Universität Freiburg)
Topology of the Heisenberg group and \(H=W\)-questions II
Approximation of Sobolev maps into Riemannian manifolds by smooth functions depends on the topology of the manifold. We report on recent progress of density questions for Sobolev maps into the Heisenberg group and related topological problems. The Heisenberg groups are examples of sub-Riemannian manifolds homeomorphic, but not diffeomorphic to the Euclidean space. Their metric is derived from curves which are only allowed to move in so-called horizontal directions.
CYCL 08, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Antonin Monteil (UCLouvain)
Méthodes métriques pour l'existence de connexions hétérocliniques en dimension infinie
Nous entendons par connexion hétéroclinique tout minimiseur du lagrangien \(\int_\mathbb{R} \frac{1}{2}|\dot{\gamma}|^2+W(\gamma) \mathrm{d} t\), et ce parmi toutes les courbes \(\gamma\) qui connectent deux puits distincts du potentiel \(W\geq 0\). Nous verrons comment établir l'existence d'une telle connexion à l'aide de la réduction standard de ce problème en un problème de géodésique : minimiser \(\int_0^1 K(\gamma) |\dot{\gamma}|\mathrm{d} t\), avec \(K=\sqrt{2W}\). Nous appliquerons cette méthode dans \(L^2\), pour résoudre des équations aux dérivées partielles stationnaires. Le cas de la dimension infinie présente d'évidentes difficultés liées au manque de compacité. Nous donnerons quelques pistes pour les résoudre, en nous focalisant sur l'existence de solutions à l'équation d'Allen−Cahn stationnaire, qui sont « hétérocliniques » en deux sens : elles connectent deux minima du potentiel à l'infini dans une direction, et deux solutions 1D stationnaires dans la direction orthogonale.
CYCL 08, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Christophe Lacave (Paris-Diderot (Paris 7))
La méthode des points vortex pour les fluides parfaits en domaine extérieur
La méthode des points vortex est une approche théorique et numérique couramment utilisée afin d'implémenter le mouvement d'un fluide parfait (en dimension deux), dans laquelle le tourbillon est approché par une somme de points vortex, de sorte que les équations d'Euler se réécrivent comme un système d'équations différentielles ordinaires. Une telle méthode n'est rigoureusement justifiée que dans le plan complet, grâce aux formules explicites de Biot−Savart. Dans un domaine extérieur, nous remplaçons également le bord imperméable par une collection de points vortex, générant une circulation autour de l'obstacle. La densité de ces points est choisie de sorte que le flot demeure tangent au bord sur certains points intermédiaires aux paires de tourbillons adjacents sur le bord. Dans ce travail, nous proposons une justification rigoureuse de cette méthode dans des domaines extérieurs. L'une des principales difficultés mathématiques étant que le noyau de Biot−Savart définit un opérateur intégral singulier lorsqu'il est restreint à une courbe.
CYCL08, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Justin Dekeyser (Université catholique de Louvain)
Desingularization of a steady point vortex in the lake equation through rearrangements
The lake equations are a particular case of 2-dimensional shallow water models, describing the averaged motion of a perfect fluid in a shallow basin with varying topography. Although the existence of solutions for the dynamical lake equations has already been proved, less has been done
for the construction of stationary solutions obtained by energy maximization. We present existence results for the problem of finding stationary solutions of the lake equations obtained by energy maximization over a class of rearrangements of a given profile-function, in the spirit of Burton's work. We prove that, as long as the profile-function asymptotically looks like a Dirac mass, so does the energy maximizer. Moreover, the concentration point
is necessarily a point of maximal depth, despite the regularity of the lake \(\Omega\). At second order, we show that the energy maximizer is sensitive to the boundary \(\partial\Omega\), and it minimizes a well-defined Kirchoff-Routh function. Furthermore, the asymptotic shape is symmetric radial nonincreasing. These results are in the spirit of Burton's and Turkington's work, although the lack of standard symmetrization techniques and the non linearity of the constraints require a different approach of the involved integral kernels.
CYCL 08, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Philippe Gravejat (Université de Cergy-Pontoise)
Asymptotic stability for solitons of the Gross−Pitaevskii and Landau−Lifshitz equations
We describe recent results about the asymptotic stability of dark solitons for the Gross-Pitaevskii and Landau-Lifshitz equations. This property is proved following the strategy introduced by Martel and Merle for the Korteweg-de Vries equation : The construction of an asymptotic profile using orbital stability, its qualitative analysis relying on monotonicity formulae, and its classification by Liouville type theorems.
This is joint work with Fabrice Béthuel (University Pierre and Marie Curie), André de Laire (University of Lille Nord de France) and Didier Smets (University Pierre and Marie Curie), and by Yakine Bahri (Nice Sophia Antipolis University).
CYCL 08, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Tadele Mengesha (University of Tennessee)
Variational convergence of some nonlocal functionals
This talk is about a class of variational problems associated with nonlocal energy functionals which result in nonlinear nonlocal systems of equations with various volumetric constraints. I will briefly discuss their derivation and connection with a nonlocal model of continuum mechanics. Next, I will show well posedness of the problem via basic variational analysis. Along the way, we will study associated energy spaces and establish connections with classical function spaces. In the event of vanishing nonlocality we establish the convergence of the nonlocal energy to a corresponding local energy via Gamma convergence. For some convex energy functionals we will explicitly find the corresponding limit energy. As a special case the classical Navier-Lame potential energy will be realized as a limit of these nonlocal energy offering a rigorous connection between the nonlocal peridynamic model to classical mechanics for small uniform strain.
CYCL08
Binh-Khoi Nguyen (Imperial College London)
Opérateurs pseudo-différentiels sur les groupes de Lie
Le calcul pseudo-différentiel est un outil standard dans l’étude des singularités d’équations aux dérivées partielles associées à des opérateurs elliptiques ou hypoelliptiques. L’importante notion sous-jacente de symbole n’étant pas invariante sous le changement de cartes locales, les opérateurs pseudo-différentiels ne peuvent être définis localement sur des variétés que si certaines conditions très strictes sur le « type » sont vérifiées. Durant l'exposé, nous considérerons une approche alternative qui consiste à utiliser la transformée de Fourier du groupe pour une définition cette fois globale des opérateurs pseudo-différentiels.
CYCL08
Alexandra Convent (UCLouvain)
Dérivées secondes faibles intrinsèques d'applications entre variétés et espaces de Sobolev d'ordre deux
Dans cet exposé, nous commençons par définir la notion de dérivée seconde pour des applications entre variétés. Cette notion permet de définir de manière intrinsèque des espaces de Sobolev d'ordre deux entre variétés. Cette nouvelle définition n'est pas toujours équivalente à la définition classique des espaces de Sobolev d'ordre deux par plongement dans un espace euclidien; pour des variétés compactes, l'espace intrinsèque est plus grand que celui par plongement. Le problème de densité des fonctions lisses est toujours ouvert. Nous donnons cependant une condition nécessaire. Nous étudions aussi une propriété de composition pour des fonctions faiblement dérivables deux fois.
Ceci est un travail en collaboration avec Jean Van Schaftingen.
CYCL09B
Christopher Hopper, Partial Regularity for Holonomic Minimisers of Quasiconvex Functionals
We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Several applications to variational problems in condensed matter physics with broken symmetries are also discussed, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals.
CYCL07 (Bâtiment Marc de Hemptinne) à Louvain-la-Neuve
Gabriele Villari (Firenze)
On the uniqueness of periodic solutions for the Liénard equation. From classical results to more recent ones.
In this talk, the problem of uniqueness of limit cycles for the Liénard system will be discussed. Starting from the well known results due to Liénard and Massera, we present some recent results which actually improve the classical ones.
CYCL03
Mircea Petrache (MPI für Mathematik Bonn)
Three types of vortices and the associated notions of optimal transport.
Vortices and topological singularities appear naturally in variational problems from physics and geometry. In particular this is the case when the function spaces which are natural for a given theory for model, contain objects (which could be vector fields, or maps, or connections on bundles) of regularity low enough that Sobolev embeddings into spaces of continuous objects do not hold. I will briefly present results concerning the classical study of Sobolev maps \(W^{1,2}(\mathbb R^3, \mathbb S^2)\), for which the model-singularity is the unit vector field \(V(x)=x/|x|\) with a singularity “of degree one” at the origin, for which the lack of approximability by smooth maps is encoded in a "minimal connection", interpretable as a discrete optimal transport plan between singularities. I then consider three other types of singularities which appear in gauge theory and for different spaces of Sobolev maps, where the replacement of the above classical discrete optimal transport is either
a) a functional on ensembles of paths having a constraint on the distribution of endpoints (partly done in projects in collaboration with L. Brasco and T. Rivière)
b) a version of discrete optimal transport “modulo 2” (studied in collaboration with R. Zuest)
c) branched transport (appearing in a recent result of F. Bethuel).
CYCL03
Pierre Bochard (Lyon)
Équation cinétique pour la selection de champs de rotationnels nul et de norme 1
Dans un article de 2002, Jabin, Otto et Perthame avaient exhibé une formulation cinétique régularisante pour certains champs de rotationnel nul à valeur dans la sphère unité de la norme euclidienne en 2d. Nous parlerons de la généralisation de ce résultat dans le cas d'une norme quelconque modulo des hypothèses de convexité. Nous montrerons en particulier comment cette formulation cinétique agit comme un principe de sélection interdisant les singularités de ligne.
CYCL 09B, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Carlo Mercuri (Swansea University)
On a class of interpolation inequalities involving Coulomb–Sobolev norms of radial functions
Café/Coffee
CYCL 09B, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Frédéric Robert (Université de Lorraine)
The Hardy–Schrödinger operator with interior singularity: mass and blow-up analysis
We consider the remaining unsettled cases in the problem of existence of positive solutions for the Dirichlet value problem \(L_\gamma u -\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}\) on a smooth bounded domain \(\Omega\) in \(\mathbb{R}^n\) (\(n\geq 3\)) having the singularity \(0\) in its interior. Here \(\gamma <\frac{(n-2)^2}{4}\), \(0\leq s <2\), \(2^*(s):=\frac{2(n-s)}{n-2}\) and \(0\leq \lambda <\lambda_1(L_\gamma)\), the latter being the first eigenvalue of the Hardy–Schrödinger operator
\[L_\gamma := -\Delta -\frac{\gamma}{|x|^2}.\]
The higher dimensional case (i.e., when \(\gamma \leq \tfrac{(n-2)^2}{4}-1\)) has been settled some time ago. In this paper we deal with the case when \(\tfrac{(n-2)^2}{4}-1<\gamma <\tfrac{(n-2)^2}{4}\). If either \(s>0\), or \(s=0\) and \(\gamma > 0\}\), we show that a solution is guaranteed by the positivity of the “Hardy-singular internal mass” of \(\Omega\), a notion that we introduce herein. On the other hand, the classical positive mass theorem is needed for when \(s=0\)\ \(\gamma \leq 0\) and \(n=3\)\, which in this case is the critical dimension.
This is joint work with Nassif Ghoussoub (University of British Columbia, Vancouver).
Repas/Lunch
CYCL 09B, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Xiao Zhong (University of Helsinki)
On the Euler–Lagrange equation of a functional by Pólya and Szegő
I will talk about a conjecture of Pólya and Szegő on minimal electrostatic capacity sets.
Café/Coffee
CYCL 09B, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Colette De Coster (Université de Valenciennes)
Existence and multiplicity results for an elliptic problem with critical growth in the gradient
In this talk, we consider the boundary value problem
\[(P_\lambda) \qquad -\Delta u = \lambda c (x) u + \mu (x) \vert \nabla u \vert^2 + h (x) u \in H^1 (\Omega) \cap L^\infty (\Omega),\]
where \(Ω \subset \mathbb{R}^N\) , \(N > 3\) is a bounded domain with smooth boundary. It is assumed that \(c\), \(h\) belong to \(L^p (\Omega)\), \(\mu\in L^\infty (\Omega)\) and \(c \gneq 0\). In case \(\lambda < 0\), we prove that the problem (P λ ) has at most one solution and we give condition under which the existence of a solution of \((P_{\lambda})\) is obtained. Moreover these solutions belong to a continuum, the behaviour of which depends in an essential way on the existence of a solution of \((P_0)\).
The geometry of the set of solutions for \(λ > 0\) is much more complicated. Under the assumption \(μ > μ_1 > 0\) for some \(μ_1 ∈ \mathbb{R}\), we derive informations on this set. In particular we prove multiplicity results as well as existence of positive, negative, non-positive, non-negative solutions. We show also that the situation is completely different for positive or negative \(h\).
D. ARCOYA , C. DE COSTER , L. JEANJEAN , K. TANAKA, Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal. 268 (2015), 2298-2335.
D. ARCOYA , C. DE COSTER, L. JEANJEAN , K. TANAKA, Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions, J. Math. Anal. Appl., 420, (2014), 772-780.
C. DE COSTER , L. JEANJEAN , Multiplicity results in the non-resonant case for an elliptic problem with critical growth in the gradient, preprint.
CYCL03
André de Laire (Lille)
The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy
It is well-known that the dynamics of biaxial ferromagnets with a strong easy-plane anisotropy is essentially governed by the Sine–Gordon equation. In this talk, we provide a rigorous justification to this observation. More precisely, we show the convergence of the solutions of the Landau-Lifshitz equation for biaxial ferromagnets towards the solutions of the Sine-Gordon equation in the regime of a strong easy-plane anisotropy, and we establish the sharpness of this convergence.
Our result holds for solutions to the Landau-Lifshitz equation in high order Sobolev spaces. We provide an alternative proof for local well-posedness in this setting by introducing high order energy quantities with better symmetrization properties. We then derive the convergence from the consistency of the Landau–Lifshitz and Sine–Gordon equations by using well-tailored energy estimates. As a by-product, we also obtain a further derivation of the free wave regime of the Landau–Lifshitz equation.
This is joint work with Philippe Gravejat (Université de Cergy-Pontoise).
local CYCL07, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Heiner Olbermann (Leipzig)
Scaling laws for thin elastic structures undergoing conical deformations
When one removes a sector from a circular sheet of paper and glues the edges back together, the resulting shape is approximately conical. Despite the simplicity of the setup, a proof of this 'fact' starting from basic models for thin elastic sheets is non-trivial. In this talk we investigate the scaling of the elastic energy with the sheet thickness h for this setting. I will show how a certain version of the Gauss curvature is the crucial quantity that controls the balance between bending and stretching of the sheet. This leads to a proof of (optimal) upper and lower bounds of the energy. As a consequence, it will be possible to prove that minimizers approach the shape of the cone as h tends to 0.
Mickaël Dos Santos (Université Paris Est Créteil Val de Marne)
Énergie renormalisée microscopique pour un modèle de Ginzburg-Landau hétérogène
Après avoir présenté et motivé la notion d'énergie renormalisée dans le cadre d'un modèle de type Ginzburg Landau, je parlerais d'un travail récent permettant d'obtenir (au moins formellement) la décomposition énergétique d’un «bébé modèle» pour l'étude d'un supraconducteur dopé (i.e. une énergie hétérogène utilisant un terme de « pinning ») par des petites impuretés.
CYCL07
Rémy Rodiac (UCLouvain)
Axially symmetric minimizers of the neo-Hookean energy in 3D
The neo-Hookean energy is an energy broadly used by physicists and engineers to describe the behavior of elastic materials undergoing large deformations. However to prove the existence of minimizers of this energy is still an open problem. We consider this problem in an axisymmetric setting and show that if the domain does not contain the axis of symmetry then minimizers do exist. Our axisymmetric minimizers are also solutions of the weak form of the Euler–Lagrange equations of elasticity.
This is a joint work with Duvan Henao (Pontificia Universidad Catolica de Chile).
Katarzina Mazowiecka (Freiburg)
Singularities of energy minimizing harmonic and minimizing biharmonic mappings from the ball to the sphere
Minimizing harmonic maps (i.e. minimizers of the Dirichlet integral) with prescribed boundary conditions from the ball to the sphere may have singularities. For some boundary data it is known that all minimizers of the energy have singularities and the energy is strictly smaller than the infimum of the energy among the continuous extensions (the so called Lavrentiev gap phenomenon occurs). We prove that the Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of boundary data. In the second part I will discuss the work in progress on biharmonic maps.
The first part of the talk is based on a joint work with P. Strzelecki.
CYCL07
Charlotte Perrin (Aix-Marseille Université)
Lagrangian approach to one-dimensional constrained systems
In this talk I will introduce and study two constrained systems which
may appear in fluid mechanics in the modelling of mixtures (constraint
on the maximal volume fraction) or of partially free surface flows
(constraint on the maximal height of the flow). I will develop a
Lagrangian approach, based on one-dimensional optimal transport tools,
which enables to obtain original existence results. I will finally show
that this approach can be also used from a numerical point of view.
CYCL07
Alessandro Fonda (Trieste)
«On the higher dimensional Poincaré−Birkhoff theorem for Hamiltonian flows»
In a joint paper with Antonio J. Ureña (Annales de l’Institut Henri Poincaré, 2017) we proposed an extension to higher dimensions of the Poincaré–Birkhoff Theorem which applies to Poincaré time-maps of Hamiltonian systems. Applications have been given to the search of periodic solutions of pendulum-type systems, weakly-coupled superlinear systems, some kind of sublinear systems, and perturbations of completely integrable systems. Recently, an extension to infinite-dimensional Hamiltonian systems has also been obtained.
Auditoire CYCL05, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Mircea Petrache (Pontificia Universidad Católica de Chile)
When does optimal transport branch?
Consider the problem of transporting some objects between \(N\) distinct locations. Depending on how we package together different objects and on how the transport cost (per unit of distance traveled) depends on the package that we are moving, we may cook up a minimum-cost transport strategy. Is it always the best option to let our objects travel independently of each other, or is it sometimes more cost-efficient to merge/split packages along the way, following a branched, tree-like, global network?
We consider the model in which the “packaging arithmetics” and the transport cost are quantified via a normed Abelian group \(G\), and we extract a purely intrinsic condition on \(G\) that guarantees that the optimal transport is not branching. This seems to initiate a new geometric classifications of certain normed groups. In the nonbranching case we also provide a global version of calibration, i.e. a generalization of Monge–Kantorovich duality.
This is joint work with Roger Züst (Universität Bern).
Auditoire CYCL05, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Bogdan Raita (Oxford)
«On some L¹–estimates»
We review some recent work on the analysis of linear elliptic systems with \(L^1\)–data. Such estimates contrast classical Calderón–Zygmund theory by Ornstein's Non-inequality. Surprising weaker estimates in full-space were proved relatively recently by Bourgain, Brezis, and Van Schaftingen, under new conditions specific to the \(L^1\)-case. We provide the analogous results on domains, which require strictly stronger conditions (joint work with F. Gmeineder). Other consequences of Van Schaftingen's Theorem will be discussed.
Auditoire CYCL05, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Katarzyna Mazowiecka (UCLouvain)
Boundary regularity of minimizing biharmonic maps
Biharmonic maps (i.e., critical points of the Hessian energy) can be seen as a generalization of harmonic maps into 4th order problems. It is known that (even minimizing) biharmonic maps may have singularities. We prove that, for sufficiently smooth boundary data minimizing biharmonic maps must be smooth in a neighborhood of the boundary.
Auditoire CYCL02, Bâtiment Marc de Hemptinne, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve
Michał Miśkiewicz (University of Warsaw)
Singularities of minimizing harmonic maps
Minimizing harmonic maps between manifolds are known to be smooth outside the so-called singular set. In general this is a rectifiable set of codimension 3, i.e., it can be covered by countably many Lipschitz pieces, but still may have many small gaps. In one special case of maps from a 4-dimensional domain into the 2-dimensional sphere Hardt and Lin proved that the singular set consists of topological curves. I will show a generalization to higher dimensional domains and discuss the topological obstruction responsible for preventing gaps in the singular set.
Salle b.203-5, Chemin du Cyclotron 2, Louvain-la-Neuve
Peter Gladbach (Universität Leipzig)
«Homogenization of transport costs»
We consider two related problems: First, the minimum cost of moving an incompressible fluid through a porous periodic medium. Second, minimizing arbitrary convex transport or flow costs on a periodic graph.
In both cases, as the length scale tends to zero, we find the homogenized limit cost, in the sense of Gamma-convergence, and discuss its relation to the porous medium equation.
This is joint work with Eva Kopfer, Jan Maas and Lorenzo Portinale.
Sagun Chanillo
Congé
Contact : Jean.VanSchaftingen@uclouvain.be