Francesca De Marchis (Università degli Studi di Roma La Sapienza)
Uniqueness via Morse index of positive solutions of the Lane-Emden problem

Abstract: We present a Morse index computation for positive solution of the classical Lane-Emden problem in planar domains, obtained via an accurate asymptotic analysis of the solutions as the exponent tends to infinity. When the domain in convex this result allows to prove uniqueness of positive solutions (for large values of the exponent), giving a first positive general answer to a conjecture raised by Dancer [JDE, 1979]. The results are obtained in collaboration with M. Grossi, I. Ianni and F. Pacella.

Riccardo Scala (Università degli Studi di Roma La Sapienza)
Some recent progress around the problem of relaxation of the area functional

Abstract: We introduce the area functional, the map which associates to any smooth function the area of its graph. We then discuss how to extend this functional to the space of general L1 functions with values in RN. The question of studying the domain and the values of the obtained extended functional has attracted much interest in the last decades since De Giorgi. In particular its behaviour when N>1 remains a big mistery and leaves several open problems, some of which posed by De Giorgi himself. We will focus on some of these open questions and discuss what kind of main issues come out when facing them. Some progress has been obtained when analyzing the area functional on piecewise constant maps.

Michiel Bertsch (Università degli Studi di Roma Tor Vergata)
Travelling wave solutions of a system of PDEs (a bit beyond FIsher-KPP)

Abstract: The structure and stability of the travelling waves of the Fisher-KPP equation are very well known. The solutions satisfy a dynamical system in the phase plane. In the seminar I shall consider a related dynamical system in 3D. Its solutions are travelling waves of a system of evolution equations. I shall discuss their structure, and present some open problems, mainly concerning their stability.

Enrique Zuazua (Universidad Autónoma de Madrid)
Population dynamics: modelling and control

Abstract: This lecture is devoted to present recent joint work in collaboration with D. Maity and M. Tucsnak (Univ. Bordeaux) on a linear system in population dynamics involving age structuring and spatial diffusion (of Lotka-McKendrick type). The model and its adjoint are of non-local nature because of the mutual effects of populations of different ages. This leads to interesting problems related to the propagation of information along the system, leading to non-standard difficulties in the frame of the control and observation of those systems. We develop a method to tackle these issues based on a combination of propagation along characteristic plus diffusion. Our results are so far limited to populations where fertility vanishes at early ages, an assumption that is not realistic for all species. We also show that controlled trajectories preserve positivity if the control time horizon is large enough.

Lucio Boccardo (Università di Roma La Sapienza)
Rileggendo la mia tesi di laurea: nuovi risultati sulla stabilità dei minimi di funzionali del Calcolo delle Variazionie

Abstract

Italo Capuzzo-Dolcetta (Università di Roma La Sapienza)
A short presentation of some results on the weak maximum principle

Abstract: I will discuss some results concerning the validity of the weak Maximum Principle wMP, that is of the sign propagation property in an unbounded domain satisfying measure-type conditions and/or geometric conditions related to the directions of ellipticity of the (possibly) degenerate elliptic fully nonlinear equation.

Andrea Marchese (Università di Pavia)
Local minimality of strictly stable extremal submanifolds

Abstract: I will discuss an extension of a result by Brian White, who proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in its homology class, if the minimization is constrained to a sufficiently small geodesic tubular neighborhood of the submanifold. We replace the tubular neighborhood with one induced by the flat distance of integral currents and we provide quantitative estimates. The proof is based on the so called "selection principle", which, via a penalization technique, allows us to recast the problem in the class of normal graphs, exploiting the regularity theory for almost minimizers. Joint work with D. Inauen (Zurich).

Samuel Nordmann (CAMS, PSL Université Paris)
Stable solutions of semilinear elliptic equations in unbounded domains

Abstract: We consider a general semilinear elliptic equation with Neumann boundary condition. A seminal result of Casten-Holland (1978) states that, if the domain is convex and bounded, all stable bounded solutions are constant. In this talk, we will investigate whether this result extends to convex unbounded domains.

Matthieu Alfaro (Université Montpellier)
Long range dispersion vs. Allee effect

Abstract: In this talk, we study the balance between long range dispersal kernels and the Allee effect in population dynamics models. To do so, we first investigate the so called Fujita blow up phenomena in presence nonlocal diffusion. We prove that the Fujita exponent dramatically depends on the behaviour of the Fourier transform of the diffusion kernel near the origin, which is linked to the tails of J. Then, as an application of the result in population dynamics models, we discuss the so called hair trigger effect. Last, if time permits, we consider the spreading properties (acceleration or not?) of equations with nonlocal diffusion and Allee effect.

Fabio Cavalletti (SISSA, Trieste)
Geometric and functional Inequalities via Optimal Transportation

Abstract: We will review some recent applications of Optimal Transportation to geometric and functional inequalities. We will review the localization technique and its application to non-smooth spaces. The quantitative version of Levy-Gromov isoperimetric inequality will also be discussed.

Luis Miguel Rodrigues Université de Rennes
Stability of traveling waves in balance laws

Abstract: The derivation from spectral stability of the asymptotic stability (in the sense of Lyapunov, i.e. in large time) of traveling waves of hyperbolic systems is an important question, that is still open to a large extent. Among difficulties to overcome stand three facts * the systems under consideration are in general quasi-linear whereas the dynamics does not exhibit strong regularization effects; * wave profiles contain in general characteristic points where, even in dimension 1, the underlying operators lose ellipticity; * wave profiles may be discontinuous so that the perturbed evolution problem becomes of mixed initial/boundary value type with free surfaces of discontinuity. In the present talk, based on recent contributions with Vincent Duchêne (CNRS, Rennes), we shall see how those difficulties may be bypassed relatively easily for waves of scalar balance laws in dimension 1.

Michela Procesi (Università degli Studi di Roma Tre)
Almost-periodic solutions for the NLS with parameters

Abstract: I shall discuss a recent result with L. Biasco and J. Massetti on the existence of almost-periodic solutions for the NLS on the circle with external parameters. After discussing the (very few) known results I shall describe our strategy, which is quite flexible and can be applied also for the construction of non maximal tori.

Isabelle Gallagher (Ecole Normale Supérieure, Paris)
Some results on the convergence from particle to Boltzmann and fluid dynamics

Abstract: In this talk we shall report on some recent progress with Thierry Bodineau, Laure Saint-Raymond and Sergio Simonella, concerning the derivation of the Boltzmann equation, and of some fluid equations, starting from particle systems as the number of particles goes to infinity, in the low density limit. We shall in particular discuss the appearance of irreversibility in the limiting procedure.

Thierry Paul (École Polytechnique, Paris)
Quantum Wasserstein

Abstract: I will define a quantum analogue to the Wasserstein distance of order two between density operators, namely positive trace one operators on Hilbert spaces. I will give first properties, in particular the link with usual Wasserstein distance between positive symbols of the quantum density operators and show how they define a kind of topology more adapted to the classical limit than the one defined by Schatten classes such as trace norm. I will present recent results obtained with E. Caglioti and F. Golse concerning a quantum version of the Brenier optimal transport Theorem and will finish by propagation estimates à la Gronwall using some quantum/classical analogue to the Wassertein metric measuring directly the "distance" between a quantum density operator and a classical probability density.

Patrick Gérard (Université Paris-Sud)
A nonlinear Fourier transform for the Benjamin-Ono equation on the circle

Abstract: The Benjamin-Ono equation is a a dispersive equation in one space dimension which admits a Lax pair involving non local operators on the Hardy space. I will report on some work in progress, in collaboration with Thomas Kappeler, devoted to the spectral theory of such operators on the circle, leading to the construction of a nonlinear Fourier transform for the Benjamin-Ono evolution.

Andrea Malchiodi (Scuola Normale Superiore, Pisa)
Prescribing Morse scalar curvatures in high dimension

Abstract: We consider the classical problem of prescribing the scalar curvature of a manifold via conformal deformation of the metric, dating back to works by Kazdan and Warner. This problem is mainly understood in low dimensions, where blow-ups of solutions are proven to be "isolated simple". We find natural conditions to guarantee this also in arbitrary dimensions, when the prescribed curvatures are Morse functions. As a consequence, we improve some pinching conditions in the literature and derive existence results of new type. This is joint work with M. Mayer. ibing Morse scalar curvatures in high dimension We consider the classical problem of prescribing the scalar curvature of a manifold via conformal deformation of the metric, dating back to works by Kazdan and Warner. This problem is mainly understood in low dimensions, where blow-ups of solutions are proven to be "isolated simple". We find natural conditions to guarantee this also in arbitrary dimensions, when the prescribed curvatures are Morse functions. As a consequence, we improve some pinching conditions in the literature and derive existence results of new type. This is joint work with M. Mayer.

Stefano Bianchini (SISSA, Trieste)
A uniqueness result for the decomposition of vector fields
Abstract

Guido De Philippis (SISSA, Trieste)
A uniqueness result for the decomposition of vector fields
Abstract: First I will review some features of the mathematical modelization of charged droplets. I will then focus on a model, proposed by Muratov and Novaga, which takes into account the regularization effect due the screening of free counterions in the droplet. In particular I will present a partial regularity result for minimizers and I will present some open problems. This is joint work with J. Hirsch e G. Vescovo.