Schedule

Abstract: We consider solutions to critical and sub-critical semilinear elliptic equations on complete, noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is non-negative, we prove both the classification of positive solutions to the critical equation and the rigidity for the ambient manifold. The same results are established for solutions to the Liouville equation on Riemannian surfaces. Our results are obtained via an appropriate P-function whose constancy implies the classification of both the solutions and the underlying manifold. The analysis carried out on the P-function also makes it possible to classify non-negative solutions for subcritical equations on manifolds enjoying a Sobolev inequality and satisfying an integrability condition on the negative part of the Ricci curvature. This is a joint work with Giulio Ciraolo e Camilla Chiara Polvara.

Abstract: We present some results about a shape optimization problem for the torsional energy associated to domains contained in an infinite cylinder, under a volume constraint. We prove that a minimizer exists for all fixed volumes, and we show some of its geometric properties. Then the question is to understand whether it is the “trivial" domain given by a bounded cylinder, whose corresponding torsion function depends only on one variable. By studying the second variation of the energy functional we are able to prove that this is not always the case. Indeed, the bounded cylinder can become unstable, which means that functions with flat level sets are not always the best candidates for optimizing the torsional energy. As the shape optimization problem is related to the variational formulation of a corresponding overdetermined problem our result also allows to deduce that, for some cylinders and some volumes, the “trivial" domain given by a bounded cylinder is not the only domain where the overdetermined problem has a solution. In fact other such domains can bifurcate from the trivial one.

The results presented are contained in papers in collaboration with D. Afonso, P. Caldiroli , A. Iacopetti, D. Ruiz and P. Sicbaldi.

Abstract: In this talk I will talk about understanding rigidity questions for four-dimensional Einstein manifolds via Harnack - type 

estimates for the Green's function of the (conformal) laplacian.   I will explain the connection of these estimates to the ADM mass of the associated AF metric

Abstract: The mathematical study of minimal surfaces, i.e. critical points of the area functional, has a long and prolific history. One of the main questions is the dimension of their singular set. The only available result in the general case is Allard’s groundbreaking work, On the first variation of a varifold, where he proved that the singular set of a stationary integral varifold is meager. This is based on an ­ɛ-regularity result for a stationary varifold close to a multiplicity one plane. Since then little to no progress has been made on the question of the optimal dimension of the singular set for integral stationary varifolds. 

Our article addresses the next interesting situation of multiplicity two, under the assumption that the stationary varifold is a 2-valued Lipschitz graph. Under these assumptions, we are able to confirm the optimal bound that the singular set is indeed of codimension one. 

In my talk, I would like to present our novelties in particular highlighting two of them. 

First, we derive an a-priori higher integrability result of the excess measure in the stationary setting. In this step the assumption Q = 2 and the graphicality are essential. But the higher integrability is crucial to be able to “pass” to a linarization. 

Second, we introduce a new class of Q-valued “gradient Young measures”. These integer rectifiable currents allow us to linearize the problem and construct Dirichlet stationary solutions. Despite their measure valued structure, we are able to establish a unique continuation result for them for every Q. Joint work with L. Spolaor.

Abstract: The stability of shear flows in the fluid mechanics is an old problem dating back to the famous Reynolds experiments in 1883. The question is to quantify the size of the basin of attraction of equilibria of the Navier-Stokes equations depending on the viscosity parameters, giving rise to the so-called stability threshold. In the case of a three-dimensional homogeneous fluid, it is known that the Couette flow has a stability threshold proportional to the viscosity, and this is sharp in view of a linear instability mechanism known as the lift-up effect. In this talk, I will explain how to exploit certain physical mechanisms to improve this bound: these can be identified with stratification (i.e. non-homogeneity in the fluid density) or rotation (i.e. Coriolis force).  Either mechanism gives rise to oscillations which suppress the lift-effect. This can be captured at the linear level in an explicit manner, and at the nonlinear level by combining sharp energy estimates with suitable dispersive estimates. 

Abstract: We will analyze the existence and the structure of different sign-changing solutions to the Yamabe equation in the whole space and we will use them to find positive solutions to critical competitive systems in dimensions 3 and 4. 

Abstract: If f is a radially symmetric function on the plane, it is elementary that its perpendicular gradient defines a stationary solution of the incompressible Euler equations, which is compactly supported if f is. In this talk we will consider the existence of nontrivial (i.e., nonradial) stationary Euler flows of compact support, and show how one can construct them using non-autonomous overdetermined boundary problems (or, more precisely, singular analogs thereof). The talk is based on joint work with Antonio Fernandez (Universidad Autonoma de Madrid) and David Ruiz (Granada). 

Abstract: I will discuss the Riemannian Penrose inequality in an asymptotically flat 3-manifold with nonnegative scalar curvature and the main points of a new proof by means of a monotonicity formula holding along the level sets of the p-capacitary potentials of the horizon of a black hole.

Joint work with Virginia Agostiniani, Lorenzo Mazzieri and Francesca Oronzio.