Let $\Omega \subset \mathbb{R}^d$ be a set with finite Lebesgue measure such that, for a fixed radius $r>0$, the Lebesgue measure of $\Omega \cap B _ r (x)$ is equal to a positive constant when $x$ varies in the essential boundary of $\Omega$. We prove that $\Omega$ is a ball (or a finite union of equal balls) provided it satisfies a nondegeneracy condition, which holds in particular for...
The curvature functionals (such as the Willmore functional) are usually defined under $W^{2,2}$ regularity assumptions on the given surface. We will explain how this assumption could be relaxed to the fractional Sobolev setting $W^{1+s, 2/s}$ for $s>1/2$, and will discuss the related problematics of a fractional variational plate model in nonlinear elasticity.
In this talk I will consider the following problem of isoperimetric type:
Given a set E in $\mathbb{R}^d$ with finite volume, is it possible to find an hyperplane $P$ that splits $E$ in two parts with equal volume, and such that the area of the cut (that is, the intersection of $P$ and $E$) is of the expected order, namely $(vol(E))^{1-1/d}$?
We can show that the answer is positive if...
In the limit of vanishing but moderate external magnetic field, we derived a few years ago together with S. Conti, F. Otto and S. Serfaty a branched transport problem from the full Ginzburg–Landau model. In this regime, the irrigated measure is the Lebesgue measure and, at least in a simplified 2d setting, it is possible to prove that the minimizer is a self-similar branching tree. In the...
We define a new rearrangement, called rearrangement by tamping, for non-negative measurable functions defined on $\mathbb{R}_+$. This rearrangement has many properties in common with the well-known Schwarz non-increasing rearrangement such as the Pólya–Szegő inequality.
Contrary to the Schwarz rearrangement, the tamping also preserves the homogeneous Dirichlet boundary condition of a...
We consider an incompressible Stokes fluid contained in a box $B$ that flows around an obstacle $K\subset B$ with a Navier boundary condition on $\partial K$. I will present existence and partial regularity results for the minimization of the drag of $K$ among all obstacles of given volume.
Together with Felix Otto, Richard Schubert, and other collaborators, we have developed two different energy-based methods to capture convergence rates and metastability of gradient flows. We will present the methods and their application to the two model problems that drove their development: the 1-d Cahn–Hilliard equation and the Mullins–Sekerka evolution. Both methods can be viewed as...
Inverse problems are about the reconstruction of an unknown physical quantity from indirect measurements. Most inverse problems of interest are ill-posed and require appropriate mathematical treatment for recovering meaningful solutions. Variational regularization is one of the main mechanisms to turn inverse problems into well-posed ones by adding prior information about the unknown quantity...
Discrete to continuum convergence results for graph-based learning have seen an increased interest in the last years. In particular, the connections between discrete machine learning and continuum partial differential equations or variational problems, lead to new insights and better algorithms.
This talk considers Lipschitz learning — which is the limit of $p$-Laplacian learning for $p$ to...
We consider an imaging inverse problem which consists in recovering a “simple” function from a set of noisy linear measurements. Our approach is variationnal: we produce an approximation of the unknown function by solving a least squares problem with a total variation regularization term. Our aim is to prove this approximation converges to the unknown function in a low noise regime....
The question of producing a foliation of the $n$-dimensional Euclidean space with $k$-dimensional submanifolds which are tangent to a prescribed $k$-dimensional simple vectorfield is part of the celebrated Frobenius theorem: a decomposition in smooth submanifolds tangent to a given vectorfield is feasible (and then the vectorfield itself is said to be integrable) if and only if the vectorfield...
We consider the Ginzburg-Landau energy $E_\epsilon$ for $\mathbb{R}^M$-valued maps defined in a cylinder $B^N\times (0,1)^n$ satisfying the degree-one vortex boundary condition on $\partial B^N\times (0,1)^n$ in dimensions $M\geq N\geq 2$ and $n\geq 1$. The aim is to study the radial symmetry of global minimizers of this variational problem. We prove the following: if $N\geq 7$, then for every...
The aim of this talk is to present results on the asymptotic analysis of a fractional version of the vectorial Allen–Cahn equation with multiple-well in arbitrary dimension. In contrast to usual Allen–Cahn equations, the Laplace operator is replaced by the fractional Laplacian as defined in Fourier space. Our results concern the singular limit $\varepsilon\to 0$ and show that arbitrary...
Let $E \to M$ be a Hermitian complex line bundle with structure group ${\rm U}(1)$ over a closed smooth orientable connected Riemannian manifold $M$. Fix a smooth metric connection ${\rm D}_0$ on $E$ and consider, for $\varepsilon > 0$, the non-self dual ${\rm U}(1)$-Yang–Mills–Higgs energies
$\displaystyle G_\varepsilon(u_\varepsilon, A_\varepsilon) := \int_M \frac{1}{2}| {\rm...
Inspired by a recent result of Lauteri and Luckhaus, we derive, via Gamma convergence, a surface tension model for polycrystals in dimension two. The starting point is a semi-discrete model accounting for the possibility of having crystal defects. The presence of defects is modelled by incompatible strain fields with quantised curl. In the limit as the lattice spacing tends to zero we obtain...
This joint work with Jean-François Babadjian is devoted to showing a discrete adaptative finite element approximation result for the isotropic two-dimensional Griffith energy arising in fracture mechanics. The problem is addressed in the geometric measure theoretic framework of generalized special functions of bounded deformation which corresponds to the natural energy space for this...
Motivated by the crystallization issue, we focus on the minimization of Heitman–Radin potential energies for configurations of $N$ particles in a periodic lattice, and in particular on the connection with anisotropic isoperimetric problems in the suitably rescaled limit as $N\to\infty$. Besides identifying the asymptotic Wulff shapes through Gamma-convergence, we obtain fluctuation estimates...
Iteratively reweighted least square (IRLS) is a popular approach to solve sparsity-enforcing regression problems in machine learning. State of the art approaches are more efficient but typically rely on specific coordinate pruning schemes. In this work, we show how a surprisingly simple reparametrization of IRLS, coupled with a bilevel resolution (instead of an alternating scheme) is able to...
In this talk, we discuss a data-driven approach to viscous fluid mechanics. Typically, in order to describe the behaviour of fluids, two different kinds of modelling assumptions are used. On the one hand, there are first principles like the balance of forces or the incompressibility condition. On the other hand there are material specific constitutive laws that describe the relation between...
I will recall the classical theory of convex duality and explain how this can be used to obtain regularity statements in the study of minimisers of the problem
$$\mathrm{min}_{u\in W^{1,p}(\Omega)}\int_\Omega F(x,\mathrm{D} u)\mathrm{d} x.$$
In particular, I will comment on recent results obtained in collaboration with Cristiana de Filippis (Parma) and Jan Kristensen (Oxford) concerning...
Entropic optimal transport (EOT) has received a lot of attention in recent years because it is related to efficient solvers. In this talk, I will address the rate of convergence of the value to the optimal transport cost as the noise parameter vanishes. This is a joint work with Paul Pegon and Luca Tamanini.
