Plenary talks

Isabeau Birindelli (Università di Roma "La Sapienza")
Title: Propagation of minima: local and non local case
Abstract: I will give a geometrical characterization of the zero level set for non negative supersolutions for operators that don't satisfy the strong maximum principle. The point of view is quite different from the classical "sharp maximum principle" of Bony. We will treat both the case of local operators and non local operators. These are joint work with Giulio Galise and Hitoshi Ishii.

Denis Bonheure (Université Libre de Bruxelles)
Title: The motion of a solid body in a Stokes or Navier-Stokes flow
Abstract: I will consider a fluid-solid model where a solid is immersed in flow constrained in a 2D infinite channel. After presenting some results on the stationary problem and the existence of local or global time-depending solutions, the focus will be put on the various open questions and challenges concerning periodic solutions. The talk is based on an ongoing research program with G. Galdi, F. Gazzola, C. Grandmond, M. Hillairet, C. Patriarca and G. Sperone.

Jean-Baptiste Casteras (Universidade de Lisboa)
Title: Almost sure well-posedness for a class of cubic Schrödinger type equations
Abstract: I will present some results concerning the local well-posedness of a cubic Schrödinger equation in R^d involving a quite general operator of order s>= 2 and with rough initial data belonging to H^S. It is well-known that there exists an explicit regularity threshold S_max(s,d) for the initial data such that if S_max(s,d) is too small then this equation is ill-posed. I will show that by randomising the initial data it is possible to prove the local well-posedness of this equation for some S below this threshold. Based on a joint work with Juraj Földes and Gennady Uraltsev.

Simão Correia (Universidade de Lisboa)
Title: Self-similar solutions and the modified Korteweg-de Vries equation
Abstract: Self-similar solutions to the modified Korteweg-de Vries equation naturally in the evolution of vortex patches, describing either sharp corners or logarithmic spirals. Moreover, they present a natural blow-up behavior at t=0 and describe the asymptotics of small solutions. However, the profile of a self-similar solution has rough decay and regularity assumptions. As such, their existence is fairly nontrivial and the dynamical flow around these solutions has to be constructed in a very delicate manner. We'll discuss the existence of self-similar solutions and their stability at both infinite and blow-up time. The results presented were obtained in collaboration with R. Cöte and L. Vega.

Cristiana De Filippis (Università di Parma)
Title: Schauder estimates for any taste
Abstract: So-called Schauder estimates are a standard tool in the analysis of linear elliptic and parabolic PDE. They have been originally obtained by Hopf (1929, interior case), and by Schauder and Caccioppoli (1934, global estimates). The nonlinear case is a more recent achievement from the '80s (Giaquinta & Giusti, Ivert, Lieberman, Manfredi). All these classical results hold in the uniformly elliptic framework. I will present the solution to the longstanding open problem (~’70s) of proving estimates of such kind in the nonuniformly elliptic setting. I will also cover the case of nondifferentiable functionals and of mixed local and nonlocal problems. From joint work with Giuseppe Mingione (University of Parma).

Manuel del Pino (University of Bath)
Title: Dynamics of concentrated vorticities in 2d and 3d
Abstract: A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on the existence and asymptotic behaviour of these solutions. We describe, with precise asymptotics, interacting vortices, and travelling helices. We rigorously establish the law of motion of "leapfrogging vortex rings", originally conjectured by Helmholtz in 1858.

Jean Dolbeault (Ceremade - Paris Dauphine)
Title: Self-similar solutions, relative entropy and applications
Abstract: Self-similar Barenblatt solutions are well known for their role in the study of large time asymptotics of the solutions of fast diffusion equations. In the framework of entropy methods, they play a key role in applications to rigidity results, symmetry or stability in interpolation inequalities. This lecture will review some results on these issues.

Fabiana Leoni (Università di Roma "La Sapienza")
Title: New concentration phenomena for sign-changing radial solutions of fully nonlinear elliptic equations
Abstract: We present recent results about radial solutions of a class of fully nonlinear elliptic Dirichlet problems posed in a ball, driven by the extremal Pucci's operators and provided with power zero order terms. In particular, we show that, for the existence of sign-changing solutions, a new critical exponent appears. Furthermore, we analyze the new concentration phenomena occurring as the exponents approach the critical values.

Gianmaria Verzini (Politecnico di Milano)
Title: Singular analysis of the optimizers of the principal eigenvalue of the Laplacian with an indefinite weight
Abstract: When analyzing the survival threshold for a species in population dynamics, one is led to consider the principal eigenvalue of some indefinite weighted problems in a bounded domain. We study the minimization of such eigenvalue, associated with Neumann boundary conditions, performing the analysis of the singular limit in case of arbitrarily small favorable region. We show that, in this regime, the favorable region is connected and it concentrates on the boundary of the domain. Though widely expected, these properties are still unknown in the general case. We also deal with some related problem, with Dirichlet boundary conditions. Joint works with Dario Mazzoleni and Benedetta Pellacci, and with Lorenzo Ferreri.