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19/06/2023 09:00
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.
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19/06/2023 10:10
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...
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19/06/2023 11:00
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...
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19/06/2023 14:00
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...
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19/06/2023 14:50
Hydrogels are crosslinked polymer networks saturated in a liquid solvent and can be modeled as
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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... -
19/06/2023 15:50
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...
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19/06/2023 16:20
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...
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20/06/2023 09:00
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...
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20/06/2023 10:10
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...
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20/06/2023 11:00
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...
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20/06/2023 14:20
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...
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20/06/2023 14:50
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...
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20/06/2023 15:50
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...
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20/06/2023 16:20
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.
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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... -
21/06/2023 09:00
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...
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21/06/2023 10:10
Given a Borel probability measure, we seek to approximate it with a measure uniformly
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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... -
21/06/2023 10:40
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...
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21/06/2023 11:10
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...
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