Dipartimento di Matematica
Guido Castelnuovo
Andrea Braides (Università di Roma Tor Vergata)
Homogenization of ferromagnetic energies on Poisson Clouds in the plane
Abstract: We prove that by scaling nearest-neighbour ferromagnetic energies defined on Poisson clouds in the plane we obtain an isotropic perimeter energy with a surface tension characterised by an asymptotic formula. The result relies on proving that cells with `very long' or `very short' edges of the corresponding Voronoi tessellation can be neglected and on computing effective metric on some random sets. Work in collaboration with A.Piatnitski.
Elena Kosygina (Baruch College)
Stochastic homogenization of viscous Hamilton-Jacobi equations with non-convex Hamiltonians: examples and open questions
Abstract: Homogenization of Hamilton-Jacobi equations with non-convex Hamiltonians in stationary ergodic random media is a largely open problem. In the last 5 years several classes of examples and counter-examples appeared in the literature. The majority of known examples concern inviscid Hamilton-Jacobi equations. We shall discuss two classes of viscous Hamilton-Jacobi equations with non-convex Hamiltonians in one space dimension for which homogenization holds and pose several open questions. The talk is based on joint works with Andrea Davini (Sapienza - Universita di Roma) and with Atilla Yilmaz (Temple University) and Ofer Zeitouni (Weizmann Institute and NYU).
Stephan Luckhaus (Universität Leipzig)
Federer, Fleming and all that
Abstract: The measure theoretic generalization of oriented submanifolds of R^N of any dimension, are currents. One the most important theorem is the compactness criterium of Federer-Fleming. We try to prove and generalize it, in collaboration with E. Spadaro, using only the theory of BV functions in R^N a' la De Giorgi.
Andrei Rodriguez (Universidad Tecnica Federico Santa Maria)
Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in R^N
Abstract: We study the behavior as t \to +\infty of unbounded solutions of the so-called viscous Hamilton-Jacobi equation in the whole space R^N, in the superquadratic case; i.e., u_t - \Delta u + |Du|^p= f(x) in R^N \times (0,+\infty), u(\cdot, 0) = u_0 in R^N for p > 2. Existence and uniqueness of viscosity solutions are shown with natural hypotheses on the initial data and right-hand side. Assuming also a growth condition on the right-hand side, we obtain what is known as ergodic large-time behavior. Joint work with Guy Barles and Alexander Quaas.
Daniele Cassani (Università dell'Insubria)
Bose fluids and positive solutions to weakly coupled elliptic system in the plane
Abstract: We prove, using variational methods, the existence in dimension two of positive vector ground states solutions for Bose-Einstein type systems. The nonlinear interaction between two Bose fluids is assumed to be of critical exponential type in the sense of J. Moser. For “small" solutions the system is asymptotically equivalent to the corresponding one with power-like nonlinearities, recently deeply studied in higher dimensions. (In collaboration with H. Tavares and J. Zhang, to appear in JDE).
Susanna Terracini (Università di Torino)
Liouville type theorems and local behaviour of solutions to degenerate or singular problems
Abstract: We consider an equation in divergence form with a singular/degenerate weight. We first study the regularity of the nodal sets of solutions in the linear case. Next, when the r.h.s. does not depend on u, under suitable regularity assumptions, we prove Hölder continuity of solutions and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the C^{0,α} and C^{1,α} a priori bounds for approximating problems Finally, we derive C^{0,α} and C^{1,α} bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.
Nicola Visciglia (Università di Pisa)
NLS: from the completely integrable to the general case
Abstract: It is well known that cubic NLS is a completely integrable model and in particular there exist infinitely many conserved quantities associated with this equation. We shall discuss a possible approach to obtain the corresponding densities which is quite flexible and can be useful to get informations in the no completely integrable case as well (for instance to get upper bounds on the polynomial growth of high order Sobolev norms). This is a joint work with F. Planchon and N. Tzvetkov.
Francesco Clemente (Sapienza Università di Roma)
Effetto regolarizzante di un termine di ordine inferiore e proprietà di regolarità locale delle soluzioni in problemi di Dirichlet ellittici non coercitivi
Abstract: In questo seminario presenterò alcuni dei risultati principali ottenuti nella mia tesi di dottorato. Essi riguardano alcune questioni relative a due classi di problemi di Dirichlet ellittici non coercitivi. Più precisamente, discuterò, per una, l'effetto regolarizzante di un termine di ordine zero di tipo polinomiale sulle soluzioni ed il comportamento asintotico di quest'ultime al tendere della potenza ad infinito; per l'altra, le proprietà di regolarità locale delle soluzioni in funzione delle proprietà di regolarità locale dei dati.
Marco Pozza (Sapienza Università di Roma)
A representation formula for viscosity solutions to PDE problems with sublinear operators
Abstract: We provide a representation formula for viscosity solutions to a nonlinear second order parabolic PDE problem with sublinear operators. This is done through a dynamic programming principle which is a generalization of the one given by Denis, Hu and Peng. It can be seen as a nonlinear extension of the Feynman-Kac formula.
Gabrielle Saller Nornberg (Sapienza Università di Roma)
On existence and multiplicity for a class of problems with natural gradient growth
Abstract: In this talk we discuss some recent results about a class of fully nonlinear second order partial differential problems in nondivergence form, uniformly elliptic with quadratic growth in the gradient. We show that multiplicity of solutions occurs up to the context of systems when the operator is not coercive, and we analyze the qualitative behavior of the continuums produced by a parameterized family of problems. Joint works with Boyan Sirakov and Delia Schiera.
Gilles Francfort (Université Paris XIII)
Homogenization for a 2D, two-phase, isotropic and periodic mixture in linear elasticity
Abstract: In this joint work with Marc Briane I will revisit one of the most elementary homogenization problems ever and demonstrate that abandonment of just one assumption, very strong ellipticity, drastically increases complexity.
Angela Pistoia (Sapienza Università di Roma)
Elliptic systems with critical growth
Abstract: I will present some results concerning the existence of nodal solutions to the Yamabe equation on the sphere and their connections with the existence of positive solutions to competitive elliptic systems with critical growth in the whole space.
Maria J. Esteban (Université Paris Dauphine)
Domains of singular Dirac operators and how to compute their eigenvalues
Abstract: In this talk I will discuss how to find and compute the eigenvalues of Dirac operators in their spectral gaps. In order to do so in an optimal way, the delicate study of the domains of critical Dirac operators is important and necessary. The results are concerned with variational methods, spectral theory and the development of optimal algorithms to compute the eigenvalues in a robust and efficient manner.
Erwin Topp Paredes (Universidad de Santiago de Chile)
Some results for the large time behavior of Hamilton-Jacobi Equations with Caputo Time Derivative
Abstract: In this talk we present Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order α ∈(0,1) cast by a Caputo derivative. The Hölder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case α=1, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions.
Ning-An Lai (Lishui University)
Strauss exponent for semilinear wave equations with scattering space dependent damping
Abstract: It is believed or conjectured that the semilinear wave equations with scattering space dependent damping admit the Strauss critical exponent. In this work, we are devoted to showing the conjecture is true at least when the decay rate of the space dependent variable coefficients before the damping is larger than 2. Also, if the nonlinear term depends only on the derivative of the solution, we may prove the upper bound of the lifespan is the same as that of the solution of the corresponding problem without damping. This shows in another way the "hyperbolicity" of the equation. This is a joint work with Z.H. Tu.
Paolo Bonicatto (Universität Basel)
Uniqueness and non-uniqueness phenomena for the transport equation in R^d
Abstract: Given d ≥ 1, T > 0 and a vector field b: [0,T] × R^d → R^d , we study the problem of uniqueness of weak solutions to the associated transport equation ∂_t u + b·∇u = 0 where u: [0,T]×R^d → R is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution to the PDE, in terms of the flow of the vector field b. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost. In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. If b is locally of class BV in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible BV vector fields and thus gives a positive answer to the (weak) Bressan’s Compactness Conjecture. The talk is based on joint works with S. Bianchini and with N.A. Gusev.
Sara Daneri (GSSI)
On the sticky particle solutions to the pressureless Euler system in general dimension
Abstract: In this talk we consider the pressureless Euler system in dimension greater than or equal to two. Several works have been devoted to the search of solutions which satisfy the following adhesion or sticky particle principle: if two particles of the fluid do not interact, then they move freely keeping constant velocity, otherwise they join with velocity given by the balance of momentum. For initial data given by a finite number of particles pointing each in a given direction, in general dimension, it is easy to show that a global sticky particle solution always exists and is unique. In dimension one, sticky particle solutions have been proved to exist and be unique. In dimension greater or equal than two, it was shown that as soon as the initial data is not concentrated on a finite number of particles, it might lead to non-existence or non-uniqueness of sticky particle solutions. In collaboration with S. Bianchini, we show that even though the sticky particle solutions are not well-posed for every measure-type initial data, there exists a comeager set of initial data in the weak topology giving rise to a unique sticky particle solution. Moreover, for any of these initial data the sticky particle solution is unique also in the larger class of dissipative solutions (where trajectories are allowed to cross) and is given by a trivial free flow concentrated on trajectories which do not intersect. In particular for such initial data there is only one dissipative solution and its dissipation is equal to zero. Thus, for a comeager set of initial data the problem of finding sticky particle solutions is well-posed, but the dynamics that one sees is trivial. Our notion of dissipative solution is lagrangian and therefore general enough to include weak and measure-valued solutions.
Andrea Corli (Università di Ferrara)
Traveling waves for degenerate parabolic equations with negative diffusivities
Abstract: In the talk I present some recent results concerning the existence and regularity of traveling waves for degenerate parabolic equations, i.e., with possibly vanishing diffusivities. Also the case of saturated diffusivities is taken into account. This study is motivated by some models of collective movements (vehicular traffic flows or crowds dynamics) and of digital treatment of images. In some cases the diffusivity can be negative.
Jean Van Schaftingen (UCLouvain)
Ginzburg-Landau functionals for a general compact vacuum manifold on planar domains
Abstract: Ginzburg–Landau type functionals provide a relaxation scheme to construct harmonic maps in the presence of topological obstructions. They arise in superconductivity models, in liquid crystal models (Landau–de Gennes functional) and in the generation of cross-fields in meshing. For a general compact manifold target space we describe the asymptotic number, type and location of singularities that arise in minimizers. We cover in particular the case where the fundamental group of the vacuum manifold in nonabelian and hence the singularities cannot be characterized univocally as elements of the fundamental group. The results unify the existing theory and cover new situations and problems. This is a joint work with Antonin Monteil (Bristol) and Rémy Rodiac (Paris–Saclay).