Schedule & Abstracts

Title:  The stability-compactness method and qualitative properties of nonlinear elliptic equations

Abstract: In this talk I report on a series of works with Cole Graham on semi-linear elliptic equations with positive non-linearities. Solutions represent stationary states of reaction-diffusion equations. We focus on qualitative properties such as uniqueness, symmetries and stability. The main motivation is to study these equations in general unbounded domains, which exhibit remarkably rich behavior. Our method rests on decomposing the problem into a compact part and one for which a stability result can be derived and then to combine the two. This approach has proved to be unexpectedly versatile and in fact encompasses past works on the subject such as the general moving plane method.

Title:  Periodic motion of a solid body in a channel flow at low Reynolds number

Abstract: I will consider a simplified fluid-solid model where an elliptical solid particle is immersed in a flow constrained in a 2D infinite channel. It is well-known that the quasi-static approximation (neglecting the inertia of the fluid and the particle) leads to a first order dynamical system which has been studied or used numerically by several authors in the community of Fluid Mechanics. Our main goal is to theoretically understand the associated phase-portrait without relying on numerics. This requires to study the properties of an elliptic solver (based on the computations of steady stokes solutions) that adjusts instantaneously the velocities of the solid by imposing that the total force and torque on the particle are zero. We rely on ideas from shape-optimization to prove the regularity of the dynamical system, on lubrication theory to show global existence of the trajectories and on asymptotic analysis to prove the stability/instability of the equilibrium positions. The quasi-static approximation can predict only two types of motions : continuous periodic rotations (tumbling) between a wall and the centreline of the channel and periodic oscillations with the long axis swinging with small-amplitude and the centre of the ellipse fluttering across the centreline. The equilibrium position with the long axis at 0 degree is unstable whereas the equilibrium position with the long axis at 90 degrees is unstable. This is a joint research with Céline Grandmont (INRIA Paris/LJLL/ULB) and Matthieu Hillairet (Univ. Montpellier).

Title:  Quasi-periodic attractors and KAM theory in Celestial Mechanics

Abstract:  The existence of quasi-periodic tori in problems of Celestial Mechanics can be obtained through Kolmogorov–Arnold–Moser (KAM) theory. Analytical proofs combined with computer-assisted techniques yield estimates in simple model problems, which turn out to be very effective. In collaboration with R. Calleja, J. Gimeno, and R. de la Llave, we studied the existence of invariant attractors in the dissipative spin-orbit problem in Celestial Mechanics, describing the rotation of a non-rigid satellite moving on a Keplerian orbit around a central planet. This problem is an example of a conformally symplectic system, which is characterized by the property to transform the symplectic form into a multiple of itself. We implemented a KAM theorem for conformally symplectic systems; finding the solution of such systems requires to add a drift parameter. The theorem gives an efficient algorithm to construct invariant attractors for the spin-orbit problem within a range of  astronomically relevant values of the parameters. It also gives accurate numerical estimates of the breakdown threshold of the invariant attractor.

Title:  Optimal partitions by isometrical sets

Abstract: I will address the question of whether there exist optimal partitions for equations in Euclidean space that have the property that all their elements are linearly isometric to each other, and I will present some results for the Schrödinger equation obtained in collaboration with Angela Pistoia (Università La Sapienza di Roma).

Title:  Liouville-type problems for the Lane-Emden equation in the half-space and cones

Abstract: In this talk, I will consider the problem of classifying the solutions, possibly unbounded and sign-changing, of the Lane-Emden equation in the half-space and cones. I will present some new Liouville-type theorems obtained in a recent joint work with L. Dupaigne and T. Petitt.

Title:  Aharonov-Bohm operators with many coalescing poles

Abstract: The behavior of simple eigenvalues of Aharonov-Bohm operators with many coalescing poles is discussed. In the case of half-integer circulation, a gauge transformation makes the problem equivalent to an eigenvalue problem for the Laplacian in a domain with straight cracks, laying along the moving directions of poles. For this problem, I present an asymptotic expansion for eigenvalues, in which the dominant term is related to the minimum of an energy functional associated with the configuration of poles and defined on a space of functions suitably jumping through the cracks.  Concerning configurations with an odd number of poles, an accurate blow-up analysis identifies the exact asymptotic behaviour of eigenvalues and the sign of the variation in some cases. An application to the special case of two poles is also discussed. The results presented in the talk have been obtained in collaboration with B. Noris, R. Ognibene, and G. Siclari.

Title: Phase space compactification for n bodies 

Abstract: For n bodies with homogeneous pair interaction we compactify every energy surface, obtaining a manifold with corners in the sense of Melrose. After a time change, the flow on it is globally defined and is non-trivial on the boundary. This is joint work with Richard Montgomery.

Title: Scattering in the classical n-body problem

Abstract: In the Newtonian n-body problem, it is unknown whether it is possible (or not) for a hyperbolic orbit to freely choose its shapes of expansion, both in the past and in the future. This is so even for the case of three equal masses on a plane! For parabolic motions, the shape of the expansion must be that of a central configuration. On the other hand, for the spatial problem of n centers, in 2017 Boscaggin Dambrosio and Terracini proved the existence of entire parabolic orbits with generic asymptotic directions for the past and the future. The goal of this talk is to present some recent advances in the investigations of this scattering problem.

Title: Hardy type inequalities involving mixed weights

Abstract: pdf file

Title:  Generalized periodic solutions in restricted three body problems

Abstract: Assume that the primaries, with masses M and m, follow elliptic orbits. The small body will satisfy a Hamiltonian system, periodic in time and having singularities at the primaries. The existence of periodic motions close to the primary M will be considered. Collisions with M will be accepted but in contrast non-resonance conditions will not be needed. The talk will discuss several results in collaboration with Alberto Boscaggin, Lei Zhao, Vivina Barutello and Gianmaria Verzini. Also, some very recent results due to Lei Zhao.

Title:   Energy stability for semilinear elliptic problems in cylinders and cones

Abstract: pdf file

Title: Chaotic dynamics and oscillatory motions in the three body problem

Abstract: Consider the planar 3 Body Problem with any masses, no all of them equal. In this talk  we address two fundamental questions: the existence of oscillatory motions and of chaotic hyperbolic sets. In 1922, Chazy classified the possible final motions of the three bodies, that is the behaviours the bodies may have when time tends to infinity. One of the possible behaviours are oscillatory motions, that is, solutions of the 3 Body Problem such that the bodies leave every bounded region but which return infinitely often to some fixed bounded region. We prove that such motions exists. We also prove that one can construct solutions of the three body problem whose forward and backward final motions are of different type. This result relies on constructing invariant sets whose dynamics is conjugated to the (infinite symbols) Bernouilli shift. These sets are hyperbolic for the symplectically reduced planar 3 Body Problem. As a consequence, we obtain the existence of chaotic motions, an infinite number of periodic orbits and positive topological entropy for the 3 Body Problem. This is a joint work with: Marcel Guardia, Pau Martin and Jaime Paradela.

Title:  Action versus energy ground states in nonlinear Schrὂdinger equations

Abstract: pdf file

Title:  Geometric variational problems: regularity vs singularity formation

Abstract: I will describe in a very informal way some techniques to deal with the existence ( and more qualitatively regularity vs singularity formation) in different geometric problems and their heat flows motivated by (variations of) the harmonic map problem, the construction of Yang-Mills connections or nematic liquid crystals. I will emphasize in particular on recent results on the construction of very fine asymptotics of blow-up solutions via a new gluing method designed for parabolic flows. I’ll describe several open problems and many possible generalizations, since the techniques are rather flexible.

Title: On the fine structure  of the two-phase free boundaries

Abstract:  pdf file

Title: The Schiffer problem on the cylinder and on the 2-sphere

Abstract: I will discuss a new result on the existence of a family of compact  subdomains  of the flat cylinder for which the Neumann eigenvalue problem for the Laplacian admits eigenfunctions with constant Dirichlet values on the boundary. These domains have the property that their boundaries have nonconstant principal curvatures. In the context of ambient Riemannian manifolds, our construction provides the first examples of such domains whose boundaries are neither homogeneous nor isoparametric hypersurfaces. The functional analytic approach we develop in this paper overcomes an inherent loss of regularity of the problem in standard function spaces. With the help of this approach, we also construct a related family of subdomains of the 2-sphere. By this we disprove a conjecture of Souam from 2005.

This is joint work with Mouhamed Moustapha Fall and Ignace Aristide Minlend.

Title: Superstability in viscoelasticity

Abstract: We consider the abstract integro-differential equation modeling the dynamics of linearly viscoelastic solids. The equation is known to generate a semigroup S(t) on a certain phase space, whose asymptotic properties have been the object of extensive studies in the last decades. In this talk we discuss some recent results obtained with F. Delloro and V. Pata on the decay rate of the semigroup compared to the decay of the memory kernel. In particular, we provide an answer to the following question: are superexponential decays possible in the context of linear viscoelasticity?

Title: A supercritical elliptic equation in the annulus

Abstract: When searching for solutions to Sobolev-supercritical elliptic problems, a major difficulty is the lack of Sobolev embeddings, that entrains a lack of compactness. In this talk, I will discuss how symmetry and monotonicity properties can help to overcome this obstacle. In particular, I will present a recent result concerning the existence of an axially symmetric solution to a semilinear elliptic equation, obtained by a combination of variational and topological techniques in the presence of invariant cones. This is a work in collaboration with A. Boscaggin, F. Colasuonno and T. Weth.

Title: Spectral partition problems with volume and inclusion constraints

Abstract: In this talk we discuss a class of spectral partition problems with a measure constraint, for partitions of a given bounded connected open set. We establish the existence of an optimal open partition, showing that the corresponding eigenfunctions are locally Lipschitz continuous, and obtain some qualitative properties for the partition. The proof uses an equivalent weak formulation that involves a minimization problem of a penalized functional where the variables are functions rather than domains, suitable deformations, blowup techniques and a monotonicity formula. Some techniques described during the talk are inspired by Susanna's work , so we will use this opportunity to celebrate and describe Susanna's creativity, talent and human qualities. The talk is based on a joint work with Ederson Moreira dos Santos (ICMC-USP),  Pêdra Andrade and Makson Santos (IST-Lisboa).

Title: Ramification of Dirichlet eigenvalues for singularly perturbed Laplace operators

Abstract: We consider Dirichlet eigenproblems for the Laplace operator in bounded domains. Suppose to perturb the problem in one of the following ways: removing a small hole from the interior of the domain, attaching a thin tube at a boundary point, disrupting the Dirichlet boundary condition with a Neumann condition in a small part of the boundary. In all these cases the operator's spectrum is stable and eigenvalues can be continued as the perturbation parameter tends to zero. We can prove the sharp asymptotic behavior of eigenbranches thanks to the so-called "Lemma on small eigenvalues" by Colin de Verdiére. In order to do this, we have to detect the quantity which can best describe the eigenvalue variation. In the first case this will be a certain notion of capacity, whereas in the second and third case it will be a certain notion of torsional rigidity. The final results show that the asymptotic behavior of eigenbranches strongly relies on the local behavior of limit eigenfunctions close to the perturbation point.

Title: Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds

Abstract: In this talk we present some results obtained jointly with Matteo Muratori (Politecnico di Milano), focusing on qualitative properties for 

• Extremals for the Sobolev inequality,

• Positive radial solutions of the Lane-Emden equation for the p-Laplacian in the critical and supercritical regimes,

• Positive radial solutions of the Lane-Emden system in the critical and supercritical regimes,

posed on a Cartan-Hadamard manifold M^n. We are particularly interested in rigidity results, both for the functions themselves, and for the underlying manifold. As an example, we show that if M^n supports an optimal function u for the Sobolev inequality, and the dimension n is less than or equal to 4, then M^n is isometric to R^n, and u is an Aubin-Talenti bubble.

Title: Spiraling Asymptotic Profiles of Competition-Diffusion Systems

Abstract: In this talk I will present a joint work with Susanna Terracini (Università di Torino) and Gianmaria Verzini (Politecnico di Milano). We study the structure of the nodal set of segregation profiles arising in the singular limit of planar, stationary, reaction-diffusion systems with strongly competitive interactions of Lotka-Volterra type, when the matrix of the interspecific competition coefficients is asymmetric and the competition parameter tends to infinity. Unlike the symmetric case, when it is known that the nodal set consists of a locally finite collection of curves meeting with equal angles at a locally finite number of singular points, the asymmetric case shows the emergence of spiraling nodal curves, still meeting at locally isolated points with finite vanishing order. I will then present a follow-up paper with Ariel Sarlot (Universidad de Buenos Aires) in which we construct solutions to the corresponding parabolic system. More specifically, we build eternal rotating spiral-like solutions.

Title:  The Hamilton–Jacobi equation on networks: from Aubry–Mather Theory to Homogenization

Abstract: Over the last few years, there has been an increasing interest in studying the Hamilton–Jacobi Equation on networks and related questions. These problems involve several subtle theoretical issues and have a significant impact on applications in various fields. While locally - i.e., on each branch of the network (arcs) - the study reduces to the analysis of 1-dimensional problems, the main difficulties arise in matching together the information converging at the juncture of two or more arcs, and relating the local analysis at a juncture with the global structure/topology of the network.

In this talk, firstly, I shall discuss several results related to the global analysis of this problem. More specifically, we developed analogs of the so-called Weak KAM theory and Aubry–Mather theory in this setting; the salient point of our approach is to associate the network with an abstract graph, encoding all of the information on the complexity of the network, and to relate the differential equation to a discrete functional equation on this graph. Then, I shall describe how to prove a Homogenization result in this context, with particular emphasis on the role of the topological complexity of the network in determining the limit problem.

Title: The number of critical points of the Robin function in domains with a small hole.

Abstract: pdf file