Focaliser sur:
Tous les jours
19 juin 2023
20 juin 2023
21 juin 2023
Indico style
Indico style - inline minutes
Indico style - numbered
Indico style - numbered + minutes
Indico Weeks View
Retour à l'affichage de la conférence
Choisissez le fuseau horaire
Utiliser le fuseau horaire de l'événement/catégorie
Veuillez spécifier un fuseau horaire
Africa/Abidjan
Africa/Accra
Africa/Addis_Ababa
Africa/Algiers
Africa/Asmara
Africa/Bamako
Africa/Bangui
Africa/Banjul
Africa/Bissau
Africa/Blantyre
Africa/Brazzaville
Africa/Bujumbura
Africa/Cairo
Africa/Casablanca
Africa/Ceuta
Africa/Conakry
Africa/Dakar
Africa/Dar_es_Salaam
Africa/Djibouti
Africa/Douala
Africa/El_Aaiun
Africa/Freetown
Africa/Gaborone
Africa/Harare
Africa/Johannesburg
Africa/Juba
Africa/Kampala
Africa/Khartoum
Africa/Kigali
Africa/Kinshasa
Africa/Lagos
Africa/Libreville
Africa/Lome
Africa/Luanda
Africa/Lubumbashi
Africa/Lusaka
Africa/Malabo
Africa/Maputo
Africa/Maseru
Africa/Mbabane
Africa/Mogadishu
Africa/Monrovia
Africa/Nairobi
Africa/Ndjamena
Africa/Niamey
Africa/Nouakchott
Africa/Ouagadougou
Africa/Porto-Novo
Africa/Sao_Tome
Africa/Tripoli
Africa/Tunis
Africa/Windhoek
America/Adak
America/Anchorage
America/Anguilla
America/Antigua
America/Araguaina
America/Argentina/Buenos_Aires
America/Argentina/Catamarca
America/Argentina/Cordoba
America/Argentina/Jujuy
America/Argentina/La_Rioja
America/Argentina/Mendoza
America/Argentina/Rio_Gallegos
America/Argentina/Salta
America/Argentina/San_Juan
America/Argentina/San_Luis
America/Argentina/Tucuman
America/Argentina/Ushuaia
America/Aruba
America/Asuncion
America/Atikokan
America/Bahia
America/Bahia_Banderas
America/Barbados
America/Belem
America/Belize
America/Blanc-Sablon
America/Boa_Vista
America/Bogota
America/Boise
America/Cambridge_Bay
America/Campo_Grande
America/Cancun
America/Caracas
America/Cayenne
America/Cayman
America/Chicago
America/Chihuahua
America/Ciudad_Juarez
America/Costa_Rica
America/Creston
America/Cuiaba
America/Curacao
America/Danmarkshavn
America/Dawson
America/Dawson_Creek
America/Denver
America/Detroit
America/Dominica
America/Edmonton
America/Eirunepe
America/El_Salvador
America/Fort_Nelson
America/Fortaleza
America/Glace_Bay
America/Goose_Bay
America/Grand_Turk
America/Grenada
America/Guadeloupe
America/Guatemala
America/Guayaquil
America/Guyana
America/Halifax
America/Havana
America/Hermosillo
America/Indiana/Indianapolis
America/Indiana/Knox
America/Indiana/Marengo
America/Indiana/Petersburg
America/Indiana/Tell_City
America/Indiana/Vevay
America/Indiana/Vincennes
America/Indiana/Winamac
America/Inuvik
America/Iqaluit
America/Jamaica
America/Juneau
America/Kentucky/Louisville
America/Kentucky/Monticello
America/Kralendijk
America/La_Paz
America/Lima
America/Los_Angeles
America/Lower_Princes
America/Maceio
America/Managua
America/Manaus
America/Marigot
America/Martinique
America/Matamoros
America/Mazatlan
America/Menominee
America/Merida
America/Metlakatla
America/Mexico_City
America/Miquelon
America/Moncton
America/Monterrey
America/Montevideo
America/Montserrat
America/Nassau
America/New_York
America/Nome
America/Noronha
America/North_Dakota/Beulah
America/North_Dakota/Center
America/North_Dakota/New_Salem
America/Nuuk
America/Ojinaga
America/Panama
America/Paramaribo
America/Phoenix
America/Port-au-Prince
America/Port_of_Spain
America/Porto_Velho
America/Puerto_Rico
America/Punta_Arenas
America/Rankin_Inlet
America/Recife
America/Regina
America/Resolute
America/Rio_Branco
America/Santarem
America/Santiago
America/Santo_Domingo
America/Sao_Paulo
America/Scoresbysund
America/Sitka
America/St_Barthelemy
America/St_Johns
America/St_Kitts
America/St_Lucia
America/St_Thomas
America/St_Vincent
America/Swift_Current
America/Tegucigalpa
America/Thule
America/Tijuana
America/Toronto
America/Tortola
America/Vancouver
America/Whitehorse
America/Winnipeg
America/Yakutat
Antarctica/Casey
Antarctica/Davis
Antarctica/DumontDUrville
Antarctica/Macquarie
Antarctica/Mawson
Antarctica/McMurdo
Antarctica/Palmer
Antarctica/Rothera
Antarctica/Syowa
Antarctica/Troll
Antarctica/Vostok
Arctic/Longyearbyen
Asia/Aden
Asia/Almaty
Asia/Amman
Asia/Anadyr
Asia/Aqtau
Asia/Aqtobe
Asia/Ashgabat
Asia/Atyrau
Asia/Baghdad
Asia/Bahrain
Asia/Baku
Asia/Bangkok
Asia/Barnaul
Asia/Beirut
Asia/Bishkek
Asia/Brunei
Asia/Chita
Asia/Choibalsan
Asia/Colombo
Asia/Damascus
Asia/Dhaka
Asia/Dili
Asia/Dubai
Asia/Dushanbe
Asia/Famagusta
Asia/Gaza
Asia/Hebron
Asia/Ho_Chi_Minh
Asia/Hong_Kong
Asia/Hovd
Asia/Irkutsk
Asia/Jakarta
Asia/Jayapura
Asia/Jerusalem
Asia/Kabul
Asia/Kamchatka
Asia/Karachi
Asia/Kathmandu
Asia/Khandyga
Asia/Kolkata
Asia/Krasnoyarsk
Asia/Kuala_Lumpur
Asia/Kuching
Asia/Kuwait
Asia/Macau
Asia/Magadan
Asia/Makassar
Asia/Manila
Asia/Muscat
Asia/Nicosia
Asia/Novokuznetsk
Asia/Novosibirsk
Asia/Omsk
Asia/Oral
Asia/Phnom_Penh
Asia/Pontianak
Asia/Pyongyang
Asia/Qatar
Asia/Qostanay
Asia/Qyzylorda
Asia/Riyadh
Asia/Sakhalin
Asia/Samarkand
Asia/Seoul
Asia/Shanghai
Asia/Singapore
Asia/Srednekolymsk
Asia/Taipei
Asia/Tashkent
Asia/Tbilisi
Asia/Tehran
Asia/Thimphu
Asia/Tokyo
Asia/Tomsk
Asia/Ulaanbaatar
Asia/Urumqi
Asia/Ust-Nera
Asia/Vientiane
Asia/Vladivostok
Asia/Yakutsk
Asia/Yangon
Asia/Yekaterinburg
Asia/Yerevan
Atlantic/Azores
Atlantic/Bermuda
Atlantic/Canary
Atlantic/Cape_Verde
Atlantic/Faroe
Atlantic/Madeira
Atlantic/Reykjavik
Atlantic/South_Georgia
Atlantic/St_Helena
Atlantic/Stanley
Australia/Adelaide
Australia/Brisbane
Australia/Broken_Hill
Australia/Darwin
Australia/Eucla
Australia/Hobart
Australia/Lindeman
Australia/Lord_Howe
Australia/Melbourne
Australia/Perth
Australia/Sydney
Canada/Atlantic
Canada/Central
Canada/Eastern
Canada/Mountain
Canada/Newfoundland
Canada/Pacific
Europe/Amsterdam
Europe/Andorra
Europe/Astrakhan
Europe/Athens
Europe/Belgrade
Europe/Berlin
Europe/Bratislava
Europe/Brussels
Europe/Bucharest
Europe/Budapest
Europe/Busingen
Europe/Chisinau
Europe/Copenhagen
Europe/Dublin
Europe/Gibraltar
Europe/Guernsey
Europe/Helsinki
Europe/Isle_of_Man
Europe/Istanbul
Europe/Jersey
Europe/Kaliningrad
Europe/Kirov
Europe/Kyiv
Europe/Lisbon
Europe/Ljubljana
Europe/London
Europe/Luxembourg
Europe/Madrid
Europe/Malta
Europe/Mariehamn
Europe/Minsk
Europe/Monaco
Europe/Moscow
Europe/Oslo
Europe/Paris
Europe/Podgorica
Europe/Prague
Europe/Riga
Europe/Rome
Europe/Samara
Europe/San_Marino
Europe/Sarajevo
Europe/Saratov
Europe/Simferopol
Europe/Skopje
Europe/Sofia
Europe/Stockholm
Europe/Tallinn
Europe/Tirane
Europe/Ulyanovsk
Europe/Vaduz
Europe/Vatican
Europe/Vienna
Europe/Vilnius
Europe/Volgograd
Europe/Warsaw
Europe/Zagreb
Europe/Zurich
GMT
Indian/Antananarivo
Indian/Chagos
Indian/Christmas
Indian/Cocos
Indian/Comoro
Indian/Kerguelen
Indian/Mahe
Indian/Maldives
Indian/Mauritius
Indian/Mayotte
Indian/Reunion
Pacific/Apia
Pacific/Auckland
Pacific/Bougainville
Pacific/Chatham
Pacific/Chuuk
Pacific/Easter
Pacific/Efate
Pacific/Fakaofo
Pacific/Fiji
Pacific/Funafuti
Pacific/Galapagos
Pacific/Gambier
Pacific/Guadalcanal
Pacific/Guam
Pacific/Honolulu
Pacific/Kanton
Pacific/Kiritimati
Pacific/Kosrae
Pacific/Kwajalein
Pacific/Majuro
Pacific/Marquesas
Pacific/Midway
Pacific/Nauru
Pacific/Niue
Pacific/Norfolk
Pacific/Noumea
Pacific/Pago_Pago
Pacific/Palau
Pacific/Pitcairn
Pacific/Pohnpei
Pacific/Port_Moresby
Pacific/Rarotonga
Pacific/Saipan
Pacific/Tahiti
Pacific/Tarawa
Pacific/Tongatapu
Pacific/Wake
Pacific/Wallis
US/Alaska
US/Arizona
US/Central
US/Eastern
US/Hawaii
US/Mountain
US/Pacific
UTC
Sauver
Europe/Paris
Français
Deutsch (Deutschland)
English (United Kingdom)
English (United States)
Español (España)
Français (France)
Italiano (Italia)
Polski (Polska)
Português (Brasil)
Türkçe (Türkiye)
Čeština (Česko)
Монгол (Монгол)
Українська (Україна)
中文 (中国)
S'authentifier
Calculus of Variations and Applications
de
lundi 19 juin 2023 (09:00)
à
mercredi 21 juin 2023 (14:00)
lundi 19 juin 2023
09:00
Sylvia Serfaty: Vortex lines in 3D Ginzburg-Landau with magnetic field
Sylvia Serfaty: Vortex lines in 3D Ginzburg-Landau with magnetic field
09:00 - 09:50
Room: Amphi 4C
In joint work with Carlos Román and Etienne Sandier, we study the onset of vortex lines in the three-dimensional Ginzburg-Landau model of superconductivity. We discuss the critical field at which the first lines appear, which is naturally connected to an "isoflux" problem. We study the optimal number of lines, their interaction, and derive a (Gamma)-limit problem for their arrangement.
09:50
Coffee break
Coffee break
09:50 - 10:10
Room: Amphi 4C
10:10
Lucia De Luca: Stability results for fractional parabolic flows
Lucia De Luca: Stability results for fractional parabolic flows
10:10 - 11:00
Room: Amphi 4C
We present an abstract method for studying the stability of parabolic flows, exploiting the Gamma-convergence of the corresponding energy functionals. We apply such a result to analyse the behavior of the s-fractional heat flows, as s tends to 0 and to 1, and of the s-Riesz flows, as s tends to 0 and to d (where d is the dimension of the ambient space). Time permitting, we present also stability results for the corresponding geometric flows.
11:00
Vincent Millot: Torus and split solutions of the Landau-de Gennes model for nematic liquid crystals
Vincent Millot: Torus and split solutions of the Landau-de Gennes model for nematic liquid crystals
11:00 - 11:50
Room: Amphi 4C
In this talk, I will present the Q-tensor model of Landau-de Gennes for nematic liquid crystals in the so called Lyutsyukov regime dealing with maps with values in the 4-dimensional sphere. This model describes stable configurations of a liquid crystal as minimizers of a Ginzburg-Landau type energy in which the potential well is the real projective plane, seen as a submanifold of S4. In the case where the 3D domain is the unit ball and the Dirichlet boundary data is radially symmetric (equivariantly), one may expect that a minimizer inherits such symmetry. Simulations show that this is not the case and a certain toroidal structure is expected to appear. If (equivariant) axial symmetry is imposed to reduce the complexity of the problem, another type of « singular » solutions appears, the split solutions. By means of regularity results on this model, I will discuss the existence / geometry of torus and split solutions and explain the strong dependence of the type of solutions with respect to the boundary condition and the shape of the domain. This talk is based on recent works in collaboration with Federico Dipasquale and Adriano Pisante.
12:00
Lunch at Restaurant Buffon
Lunch at Restaurant Buffon
12:00 - 14:00
Room: Amphi 4C
14:00
Sergio Conti: Laminates with variable volume fraction in shape-memory alloys
Sergio Conti: Laminates with variable volume fraction in shape-memory alloys
14:00 - 14:50
Room: Amphi 4C
I will discuss a singularly perturbed variational model for single laminates in shape-memory alloys, with boundary conditions that induce a position-dependent volume fraction. The scaling of the minimum value of the (geometrically linear) energy with respect to the surface energy density is determined by an explicit upper bound and an ansatz-free lower bound, both for a Dirichlet and for a Neumann problem. The lower bound builds upon a rigidity estimate for functions of bounded deformation. This talk is based on joint work with R. V. Kohn and O. Misiats.
14:50
Leonie Schmeller: Gel models for phase separation at finite strains
Leonie Schmeller: Gel models for phase separation at finite strains
14:50 - 15:20
Room: Amphi 4C
Hydrogels are crosslinked polymer networks saturated in a liquid solvent and can be modeled as a two-phase system employing the phase field approach. During swelling and squeezing, they un- dergo enormous volume changes, which requires finite strain models for realistic considerations. We analytically investigate the two-phase model for phase separation in a geometrically nonlinear elas- tic material. The coupled system of PDEs consists of a Cahn-Hilliard equation and a quasi-static mechanical force balance of the deforming gel. The phase field and the mechanics are coupled by a multiplicative decomposition of the deformation gradient, and time-dependent Dirichlet boundary conditions are imposed on the deformation field. Based on a time-incremental scheme, we derive existence theory of solutions in a suitable weak formulation. Using techniques from the calculus of variations and nonlinear PDE theory, we obtain further an existence result for the time-continuous problem under suitable assumptions. This is a joint work with Marita Thomas within the DFG priority program SPP 2171 Dynamic wetting of flexible, adaptive and switchable substrates, project \# 422786086 and within the MATH+ project AA2-9.
15:20
Coffee break
Coffee break
15:20 - 15:50
Room: Amphi 4C
15:50
Andreas Vikelis: Measure-valued solutions for non-associative finite plasticity
Andreas Vikelis: Measure-valued solutions for non-associative finite plasticity
15:50 - 16:20
Room: Amphi 4C
The variational treatment of evolutionary non-associative elasto-plasticity at finite strains remains unexplored. In this direction, following the concept of energetic solutions, we present an existence result for measure-valued solutions of the quasistatic evolution problem which are stable and balance the energy. In particular, we apply a modification of the standard time-discretization scheme, considering Young measures generated by piecewise constant interpolants of time-discrete solutions of a properly defined minimization problem. A key point in our analysis is the limit passage in the dissipation energy. The later calls for time-continuity properties of the stresses which are not expected in the quasistatic framework. To overcome this obstacle we introduce a regularization of the generalized stress in the definition of our energetic solutions. Joint work with Ulisse Stefanelli.
16:20
Barbara Zwicknagl: Variational models for pattern formation: from helimagnets to shape-memory alloys
Barbara Zwicknagl: Variational models for pattern formation: from helimagnets to shape-memory alloys
16:20 - 17:10
Room: Amphi 4C
We consider continuum variational models for pattern formation in helimagnetic compounds. The energy functional consists of a multi-well bulk energy regularized by a higher order interfacial energy, and arises from a frustrated spin model in the sense of Gamma-convergence. We derive the scaling law for the minimal energy in the case of incompatible boundary conditions. The scaling law indicates the formation of various branching-type patterns in certain parameter regimes. We in particular outline relations to well-studied variational models for martensitic microstructures. This talk is based on joint works with Janusz Ginster and Melanie Koser (both Humboldt-Universität zu Berlin).
mardi 20 juin 2023
09:00
Simone Di Marino: The shape of Kantorovich potentials: on the existence of minimizers for the Lieb-Oxford inequality
Simone Di Marino: The shape of Kantorovich potentials: on the existence of minimizers for the Lieb-Oxford inequality
09:00 - 09:50
Room: Amphi 4C
We explain the connection between the classical Lieb-Oxford inequality and multimarginal optimal transport with repulsive cost. We can see that the first order condition is linked with the Kantorovich potential, and we show, through a detailed analysis of the shape of the potentials, that if a minimizer exists, then it should be compactly supported, extending the case N=1 which was already settled by Lieb and Oxford in their original contribution. This is a work in preparation with R. Lelotte (U. Paris-Dauphine)
09:50
Coffee break
Coffee break
09:50 - 10:10
Room: Amphi 4C
10:10
Manuel Friedrich: Equilibrium configurations for epitaxially strained crystalline films
Manuel Friedrich: Equilibrium configurations for epitaxially strained crystalline films
10:10 - 11:00
Room: Amphi 4C
In this talk, we revisit results obtained on the existence of minimizers and relaxation for energies related to epitaxially strained crystalline films. We first extend the analysis to the framework of three-dimensional linear elasticity. Afterwards, we discuss a rigorous relation between models in nonlinear and linearized elasticity for both continuum and atomistic energies. Based on joint works with Vito Crismale, Leonard Kreutz, and Konstantinos Zemas.
11:00
Paul Pegon: Asymptotics for optimal quantization in branched optimal transport
Paul Pegon: Asymptotics for optimal quantization in branched optimal transport
11:00 - 11:50
Room: Amphi 4C
The problem of optimal quantization of measures consists in finding the best approximation of a given measure by an atomic measure with a fixed number of atoms, usually expressed through Wasserstein distances. One can formulate the same problem considering instead the irrigation distances of branched optimal transport, where the transport cost behaves as a concave power of the mass and depends on all the trajectories of the particles. We study the asymptotic behaviour of optimal quantizers for absolutely continuous measures as the number of atoms grows to infinity. We compute the limit distribution of the corresponding point clouds and show in particular a branched transport version of Zador's theorem. Moreover, we establish the asymptotic quasi-uniformity of optimal quantizers in terms of separation distance and covering radius of the atoms, when the measure is uniform. This is a joint work with Mircea Petrache.
12:00
Lunch at Restaurant Buffon
Lunch at Restaurant Buffon
12:00 - 14:20
Room: Amphi 4C
14:20
Liangjun Weng: A constrained mean curvature type flow and isoperimetric type inequalities
Liangjun Weng: A constrained mean curvature type flow and isoperimetric type inequalities
14:20 - 14:50
Room: Amphi 4C
In this talk, we will discuss the isoperimetric inequality and its high-order version -- Alexandrov-Fenchel inequality, which dates back to Queen Dido in the ancient Carthage era. We introduce the quermassintegrals for compact hypersurfaces with capillary boundary from the variational viewpoint. Then by using a constrained mean curvature type flow, we can obtain some new isoperimetric type inequalities for compact hypersurfaces with capillary boundary.
14:50
Eloi Martinet: Numerical maximization of Neumann eigenvalues on domains on the sphere
Eloi Martinet: Numerical maximization of Neumann eigenvalues on domains on the sphere
14:50 - 15:20
Room: Amphi 4C
We consider the numerical optimization of the first three eigenvalues of the Laplace-Beltrami operator of domains on the sphere with Neumann boundary conditions. We adress two approaches : one is a shape optimization procedure via the level-set method and the other one is a relaxation of the initial problem leading to a density method. These computation gives some strong insight on the optimal shapes of those eigenvalue problems and shows a rich variety of shapes regarding the proportion of the surface area of the sphere occupied by the domain.
15:20
Coffee break
Coffee break
15:20 - 15:50
Room: Amphi 4C
15:50
Alice Marveggio: Uniqueness and stability of planar multiphase mean curvature flow beyond a circular topology change
Alice Marveggio: Uniqueness and stability of planar multiphase mean curvature flow beyond a circular topology change
15:50 - 16:20
Room: Amphi 4C
The evolution of a network of interfaces by mean curvature flow features the occurrence of topology changes and geometric singularities. As a consequence, classical solution concepts for mean curvature flow are in general limited to short-time existence theorems, which include singular times only for some stable shrinkers such as the circle. At the same time, the transition from strong to weak solution concepts (e.g. Brakke solutions) may lead to non-uniqueness of solutions. Following the relative energy approach à la Fischer-Hensel-Laux-Simon and introducing a suitable notion of gradient-flow calibration for a shrinking circle, we prove a quantitative stability estimate holding up to the singular time. This implies a weak-strong uniqueness principle for weak BV solutions to planar multiphase mean curvature flow beyond circular topology changes. Furthermore, we expect our method to have further applications to other types of shrinkers, as well as to prove quantitative convergence of diffuse-interface (Allen-Cahn) approximations for mean curvature flow. This is work in progress with Julian Fischer, Sebastian Hensel and Maximilian Moser.
16:20
Tim Laux: The large-data limit of the MBO scheme for data clustering
Tim Laux: The large-data limit of the MBO scheme for data clustering
16:20 - 17:10
Room: Amphi 4C
The MBO scheme is an efficient algorithm for data clustering, the task of partitioning a given dataset into several meaningful clusters. In this talk, I will present the first rigorous analysis of this scheme in the large-data limit. The starting point for the first part of the talk is that each iteration of the MBO scheme corresponds to one step of implicit gradient descent for the thresholding energy on the similarity graph of the dataset. It is then natural to think that outcomes of the MBO scheme are (local) minimizers of this energy. We prove that the algorithm is consistent, in the sense that these (local) minimizers converge to (local) minimizers of a suitably weighted optimal partition problem. To study the dynamics of the scheme, we use the theory of viscosity solutions. The main ingredients are (i) a new abstract convergence result based on quantitative estimates for heat operators and (ii) the derivation of these estimates in the setting of random geometric graphs. To implement the scheme in practice, two important parameters are the number of eigenvalues for computing the heat operator and the step size of the scheme. Our results give a theoretical justification for the choice of these parameters in relation to sample size and interaction width. This is joint work with Jona Lelmi (U Bonn).
18:45
Social Dinner at Bouillon République
Social Dinner at Bouillon République
18:45 - 20:45
Room: Amphi 4C
mercredi 21 juin 2023
09:00
Maria Giovanna Mora: Explicit minimizers for anisotropic Coulomb energies
Maria Giovanna Mora: Explicit minimizers for anisotropic Coulomb energies
09:00 - 09:50
Room: Amphi 4C
Nonlocal interaction energies play a pivotal role in describing the behavior of large systems of particles, in a variety of applications. Traditionally, the focus of the mathematical literature on nonlocal energies has been on radially symmetric potentials, which model interactions depending on the mutual distance between particles. The mathematical study of anisotropic potentials, despite their natural occurrence in modeling interactions where a preferred direction of interaction is present, has on the other hand been very limited until recently. In this talk we will consider a general class of anisotropic energies of Coulomb type in three dimensions and give a complete characterization of their minimizers, under the sole assumption of non-negativity for the Fourier transform of the interaction kernel.
09:50
Coffee break
Coffee break
09:50 - 10:10
Room: Amphi 4C
10:10
Joao Machado: 1D approximation of measures in Wasserstein spaces
Joao Machado: 1D approximation of measures in Wasserstein spaces
10:10 - 10:40
Room: Amphi 4C
Given a Borel probability measure, we seek to approximate it with a measure uniformly distributed over a $1$-dimensional set. With this end, we minimize the Wasserstein distance of this fixed measure to all probability measures uniformly distributed to connected $1$ dimensional sets and a regularization term given by their length. To show existence of solution to this problem, one cannot easily resort to the direct method in the calculus of variations due to concentration of mass effects. Therefore, we propose a relaxed problem in the space of probability measures which always admits a solution. In the sequel, we show that whenever the initial measure has $L^1$ density w.r.t. the $1$-Hausdorff measure (in particular for absolutely continuous measures w.r.t. Lebesgue) then the solution will be a rectiable measure. This allows us to perform a blow-up argument that, in dimension $2$, shows that the solution has a uniform density, being therefore a solution to the original problem. Finally, in any dimension, we manage to prove that solutions to the relaxed problem are Ahlfors regular.
10:40
Annette Dumas: Existence and Lipschitz regularity of the trajectories minimizing the total variation in a congested setting
Annette Dumas: Existence and Lipschitz regularity of the trajectories minimizing the total variation in a congested setting
10:40 - 11:10
Room: Amphi 4C
The problem I will present is motivated by the study of a Mean Field Game model whose theory was simultaneously introduced by Lasry and Lions and by Caines, Huang and Malhamé in 2006. The model consists in studying a population in a city where each agent jumps to move from one place to another. Each inhabitant minimizes a cost composed of the number of jumps and an increasing function of the density of the population. The solution to this problem is a probability measure on the trajectories which is a Nash equilibrium. The probability measure Q on the trajectories can be seen as a trajectory of the density of the population which leads us to the minimization of a variational problem which depends on the $L^1$-norm of the speed of the density which is linked to the number of jumps and the additional cost which is associated with the increasing function of the density. Density constraints are also added to the problem. We will see that the solution exists, is unique and is Lipschitz in time, despite the discontinuous trajectories taken by each agent. With additional hypothesis on the data, boundedness or continuity in space can be obtained with Dirichlet conditions in time. The aspect of the solutions are given by the Euler-Lagrange equations which show that in space, either the solution is constant, or it follows the critical points of the cost. Numerical simulations are carried out on a simple example by using the fast dual proximal gradient method from Beck which validates the theoretical framework.
11:10
Antoine Lemenant: Epsilon-regularity for Griffith minimizers
Antoine Lemenant: Epsilon-regularity for Griffith minimizers
11:10 - 12:00
Room: Amphi 4C
In this talk I will present some recent advances concerning the $C^1$ regularity of minimizers for the vectorial free-discontinuity problem of Griffith. In particular I will try to explain the strategy of proof inspired by the Reifenberg-flat theory, relying on a geometric stopping time argument on the flatness, coupled with a general extension lemma, which was employed in our latest result valid for any dimension $N>2$. This is a recent joint work with C. Labourie, and generalizes, with a different proof, a previous 2 dimensional result obtained in collobaration with J.F. Babadjian and F. Iurlano.
12:00
Lunch at Restaurant Buffon
Lunch at Restaurant Buffon
12:00 - 14:00
Room: Amphi 4C