Abstract We consider second-order ergodic Mean-Field Games systems in \(\mathbb{R}^N\) with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. Equilibria solve a system of PDEs where an Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution.
Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for existence and nonexistence of classical solutions to the MFG system. In the Hardy-Littlewood-Sobolev-supercritical regime, by means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential term. On the other hand, in the Hardy-Littlewood-Sobolev-subcritical regime, using a fixed point argument, we show existence of classical solutions at least for masses smaller than a given threshold value. In the mass-subcritical regime we show that actually this threshold can be taken to be \(+\infty\).
Finally, considering the MFG system with a small parameter \(\varepsilon>0\) in front of the Laplacian, we study the behavior of solutions in the vanishing viscosity limit, namely when the diffusion becomes negligible. First, we obtain existence of classical solutions to potential free MFG systems with Riesz-type coupling. Secondly, we prove concentration of mass around minima of the potential.
This talk is based on a joint work with A. Cesaroni (Bernardini C., Cesaroni A.: Ergodic Mean-Field Games with aggregation of Choquard-type, J. Differential Equations 364, 296-335 (2023). Preprint ArXiv: https://arxiv.org/abs/2208.08177) and on the work Bernardini C.: Mass concentration for Ergodic Choquard Mean-Field Games, (2022) submitted, ArXiv: https://arxiv.org/abs/2212.00132.
Abstract Recently, we studied the Legendre-Hardy inequality, a Neumann version of Hardy's inequality. In this talk, I would like to introduce an extension of the Legendre-Hardy inequality to the Legrendre-Hardy-Sobolev inequality, which combining the Legendre-Hardy inequality and the Sobolev inequality.
Abstract We show that the circular rearrangement does not increase the \(L^p\) norm of the gradient of a Sobolev function. Our analysis can be applied to functionals whose integrand is a convex function of the gradient, and to functions that do not necessarily satisfy Dirichlet boundary conditions. This generalises previous contributions by Pólya and several authors. After that, we study rigidity of the inequality. That is, we discuss under which conditions all the extremals of the inequality are symmetric.
This is work in collaboration with Georgios Domazakis (University of Sussex), Matteo Perugini (University of Milan), and Francis Seuffert.
Abstract The horizontal mean curvature flow is widely used in neurogeometry and in image processing (e.g. Citti-Sarti model). It represents, informally, the contracting evolution of a hypersurface embedded in a particular geometrical setting, called sub-Riemannian geometry, in which only some curves (called horizontal curves) are admissible by definition. This may lead the existence of some points of the hypersurface, called characteristic, in which is not possible to define the horizontal normal. In order to avoid this problem, it is possible to use the notion of Riemannian approximation of a sub-Riemannian geometry applied to the horizontal mean curvature flow.
I will show the connection between the evolution of a generic hypersurface in this setting and the associated stochastic optimal control problem. Then, I will show some results asymptotic optimal controls in the Heisenberg group and use them to later show a convergence result between the solutions of the approximated mean curvature flow and the horizontal ones. This is from some joint works with N. Dirr and F. Dragoni.
Abstract We study existence and qualitative properties of the minimizers for a Thomas-Fermi type energy functional defined by \[ \mathcal{E}_\alpha(\rho) := \frac{1}{q} \int_{\mathbb{R}^d} |\rho(x)|^q \, dx + \frac{1}{2} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{\rho(x) \rho(y)}{|x - y|^{d - \alpha}} \, dx \, dy - \int_{\mathbb{R}^d} V(x) \rho(x) \, dx, \] where \(d \ge 2\), \(\alpha \in (0, d)\) and \(V\) is a potential. Under broad assumptions on \(V\) we establish existence, uniqueness and qualitative properties such as positivity, regularity and decay at infinity of the global minimizer. The decay at infinity depends in a nontrivial way on the choice of \(\alpha\) and \(q\). We also show that under some conditions on \(V\) the global minimizer is sign-changing even if \(V\) is nonnegative. In such regimes we establish a relation between the positive part of the global minimizer and support of the minimizer of the energy, constrained on the non-negative functions. Our study is motivated by some recent models for charge screening in graphene, where sign-changing minimizers appear in a natural way.
Abstract This talk is concerned with the Dirichlet eigenvalue problem associated to the infinity Laplacian in metric spaces. We provide a PDE approach to find the principal eigenvalue and eigenfunctions for a bounded domain in a proper geodesic space with no measure structure. We give an appropriate notion of solutions to the infinity eigenvalue problem and show the existence of solutions by adapting Perron's method. Our method is different from the standard limit process, introduced by Juutinen, Lindqvist and Manfredi in 1999, via the variational eigenvalue formulation for p-Laplacian in the Euclidean space. Several further results and concrete examples will be given in the case of finite metric graphs. This talk is based on joint work with Ayato Mitsuishi at Fukuoka University.
Abstract The Mean Field Games (MFGs) model describes interactions among a very large number of identical agents. Evolutive first order MFGs occurs when the time horizon is finite, and the dynamics of the agents are deterministic; they are modelized by a system of two coupled equation: a Hamilton- Jacobi equation and a continuity equation describing respectively the optimal cost of a generic agent and the distribution of the whole population. I will talk about a joint research project with Y. Achdou, C. Marchi and N. Tchou about some models of MFGs where the Hamiltonian is not coercive in the gradient term because the dynamics of the generic player must fulfil some constraints or fail to be controllable. First of all, I will outline the model where the generic player can move in the whole space, but it has some forbidden directions. Afterwards, I will treat the case where the dynamics of the generic agent are controlled by the acceleration. We study the existence of weak solutions, and we relate it with the relaxed equilibria in the Lagrangian setting which are described by a probability measure on the set of optimal trajectories.
Abstract
We study singular perturbation problems for second order Hamilton-Jacobi equations in unbounded spaces. In these problems the differential operator depends on a
parameter epsilon and splits in two parts: one which varies faster and faster as epsilon tends to 0 and depends only on some variables (so-called fast
variables) and one which varies slowly and depends on the other variables (so-called slow
variables). In our problems, the fast operator is linear, uniformly
elliptic and has an Ornstein-Uhlenbeck type drift (no periodicity is required) while the slow operator is a fully nonlinear (possibly degenerate) elliptic operator,
and the source term is only locally Hölder continuous. We prove that, as epsilon tends to 0, the solution to these problems converges to the solution of a
suitable differential equation in the sole slow variables where the contribution of the fast operator is somehow homogenized. Mainly, we shall provide several rates
of convergence according to the regularity of the source term.
This is a joint work with Daria Ghilli (University of Pavia).
Abstract I will discuss a class of quasilinear elliptic equations involving the p-Laplace operator and nonlinearities of Sobolev-critical growth, focusing on existence, non-existence, and compactness issues related to their variational formulation.
Abstract We discuss the asymptotic behaviour of groundstates for a class of singularly perturbed Choquard equations with a local repulsion term. We identify seven different asymptotic regimes and provide a characterization of the limit profiles of the groundstates when perturbation parameter is small. We also outline the behaviour of groundstates when perturbation is strong. In some of the regimes the limit profile is given by a compactly supported discontinuous minimizer of a Thomas-Fermi type variational problem.
This is joint work with Zeng Liu (Suzhou, China) and Damiano Greco (Swansea).
Abstract Consider a function \(u \colon \mathbb{R}^n \to \mathbb{R}\) such that its gradient is of locally bounded variation and takes values in a finite set \(A \subset \mathbb{R}^n\) almost everywhere. Thus the preimages \(\{x \in \mathbb{R}^n \colon \nabla u(x) = a\}\), for \(a \in A\), form (locally) a finite Caccioppoli partition. Such functions can arise in certain models from materials science, for example in the context of an Allen-Cahn (Modica-Mortola) type energy involving the gradient.
The theory of Caccioppoli partitions gives some information about the structure of \(u\). The fact that we are dealing with a gradient, however, gives rise to much more rigidity, especially if the set \(A\) is convex independent or even affine independent. Then \(u\) is piecewise affine away from a small subset of the domain.
Abstract Considering the Navier-Stokes equations, we quantitatively investigate the behaviour of potentially singular solutions. Inspired by a Calderon splitting argument, we produce bounds of solutions of the Navier Stokes equations in terms of norms of the initial data in rough function spaces. We also look at applications of these bounds concerning the quantitative properties of the (potential) singular set.
Abstract
The strong relationship between log-concavity and the heat flow is well known.
In particular, it is known that the log-concavity of the initial datum is preserved
by the Dirichlet heat flow and that, starting with a compactly supported initial datum,
log-concavity of the solution eventually appears after a sufficiently long time.
I will present the results of a joint work with Kazuhiro Ishige and Asuka Takatsu where we
investigate deeply this strong relationship and we introduce a different kind of concavity property
which is somehow optimal
for the Dirichlet heat flow.
Abstract The Alexandrov-Fenchel (AF) inequalities are generalizations of Isoperimetric inequalities. In this presentation, I will discuss the stability of AF inequalities in the hyperbolic space. To understand what I mean by stability, I will first explain it in the context of the Isoperimetric inequality.
Abstract Nonlocal energies are continuum models for large systems of particles with long-range interactions. Under the assumption that the interaction potential is radially symmetric, several authors have investigated qualitative properties of energy minimisers. But what can be said in the case of anisotropic kernels?
I will present some results and partial answers in this direction obtained in a long-standing collaboration with Maria Giovanna Mora and Luca Rondi, and with Jose Antonio Carrillo, Joan Mateu and Joan Verdera.
Abstract In this work we investigate existence of solutions for a stress-rate type model of viscoelastic material response in the context of strain-limiting theory. First, we use a stress-rate type constitutive relation and the equation of motion to derive a partial differential equation where unlike classical equations in elasticity the unknown is the stress rather than the deformation. Secondly, we introduce a variational framework to prove existence of solutions to this equation by considering it as the Euler-Lagrange equation corresponding to a suitably chosen functional. We use the method of minimisation of this functional in a time-discretized setting in order to solve the problem. This is a joint work with Anja Schlomerkemper and Luisa Bachmann, both from the University of Wurzburg.
Abstract I will present a result obtained in collaboration with Apala Majumdar, Giacomo Canevari and Yiwei Wang. Our setting is a nematic liquid crystal (NLC) that occupies a region \(\Omega\) (open set, simply connected) in the plane. We are working with dilute ferronematics suspensions, that is magnetic nanoparticles inside the NLC such that the distance between pairs of MNPs is much larger than the individual MNP sizes and the volume fraction of the MNPs is small. In these dilute systems, the MNP-MNP interactions and the MNP-NLC interactions are absorbed by an empirical magneto-nematic coupling energy. Our energy has three components: a reduced LdG free energy for NLCs, a Ginzburg-Landau free energy for the magnetization and a homogenised magneto-nematic coupling term. Here \(\mathbf{Q} \colon\Omega \to \mathcal{S}^{2\times2}_0\) is a \(\mathbf{Q}\) tensor and \(\mathbf{M}\colon \Omega\to \mathbb{R}^2\) is the magnetic field. \begin{equation} \tag{1} \mathcal{F}_{\epsilon} (\mathbf{Q}, M) := \int_{\Omega} \left(\frac12 |\nabla \mathbf{Q}|^2 + \frac{\xi}2 |\nabla \mathbf{M}|^2 + \frac1{\epsilon^2} f(\mathbf{Q}, \mathbf{M})\right) \mathrm{d} x \end{equation}
In two dimensions, we have
\[
f(\mathbf{Q}, \mathbf{M}) := \frac14 (|\mathbf{Q}|^2-1)^2 + \frac{\xi}4 (|\mathbf{M}|^2-1)^2 - c_0 \mathbf{Q} \mathbf{M}\cdot \mathbf{M} , \quad c_0>0.
\]
We study a special limit of the effective free energy in (1), for which both \(\xi\) and \(c_0\) are proportional to \(\epsilon\)
and we study the profile of the corresponding energy minimizers in the \(\epsilon \to 0\) limit, subject to Dirichlet boundary conditions for
\(\mathbf{Q}\) and \(\mathbf{M}\). This can be interpreted as a super-dilute
limit of the ferronematic free energy for which the magnetic energy is substantially
weaker than the NLC energy, and the magneto-nematic coupling is weak.
Abstract For Liouville equation with quantized singular sources, the nonsimple blowup phenomenon has been a major difficulty for years. In this talk I will report on several progress we made in the last few years. First we prove that for the classical Liouville equation \[ \Delta u+\lambda e^u = 4\pi \sum_{j=1}^m \gamma_j \delta_{p_j}, \mbox{in} \ \Omega; u=0 \ \mbox{on} \ \partial \Omega \] all blow-ups are simple, even in the quantitized case. Secondly we prove that non-simple blow-ups, if exist, must be single, i.e. no other blow-ups can co-exists with non-simple blow-up. Finally we prove some vanishing theorems up to the second order derivatives.