Seminari in programmazione durante l'anno accademico 2011- 2012
Valentino Magnani
(Università di Pisa):
Stime integrali per funzioni convesse rispetto a campi di
Hormander.
(martedì 5 giugno 2012, ore 12,15, sala riunioni VII piano).
Sunto. Presenteremo alcuni risultati di regolarità per funzioni convesse
rispetto a campi vettoriali che soddisfano la condizione di Hormander.
Andrea Pinamonti
(Università di Padova):
Titolo da definire
(martedì 12 giugno 2012, ore 12,15 aula 1BC45).
Seminari A.A. 2011-2012 (dai più recenti ai più vecchi)
Lawrence C. Evans
(University of California, Berkeley):
Adjoint methods for Nonlinear PDE
(7-18 maggio 2012).
Nell'ambito del Dottorato in Matematica.
Diogo Gomes
(Universidade Tecnica de Lisboa):
A-priori estimates for mean-field games
(15 maggio 2012).
Abstract: In this talk we discuss a number of techniques to obtain a-priori estimates for stationary mean-field games. First we explore the special variational structure of a class of mean-field games to obtain various regularity results by elementary methods. Then we use the Evans adjoint method to obtain a-priori regularity estimates for viscosity solutions
of Hamilton-Jacobi equations with Sobolev coefficients. This allow us to establish the existence of smooth solutions for a class of stationary mean-field games.
Nikolay Pogodaev
(Università di Brescia):
Control Problems in Individuals-Population Interactions
(26 aprile 2012).
Abstract:
This seminar is devoted to a model for individuals-population
interactions based on differential inclusions. Within this framework,
we consider new control problems where the state to be controlled is a
set. In particular, we aim at confining the region occupied by the
population. Rigorous results about finite/infinite time confinement
and impossibility of confinement are presented. A wide variety of
analytical tools are exploited: from quasidifferential equations in
metric space to measure theory, from non-smooth analysis to classical
o.d.e. techniques. Finally, numerical integrations of sample
situations are used to illustrate further open problems.
Geometric PDEs and applications
Workshop
Organizers: Annalisa Cesaroni, Matteo Novaga, Luca Rossi
(19-20 aprile 2012).
Khai T. Nguyen
(Università di Padova):
Lower compactness estimates for scalar balance laws
(12 aprile 2012).
Abstract
Incontro "Nonlinear PDEs",
Lo scopo di quest' incontro e' stato quello di parlare, in maniera breve e informale, degli argomenti attuali di ricerca tra i partecipanti al seminario di Equazioni Differenziali e Applicazioni.
Interventi di 15 minuti e di carattere introduttivo.
(21 marzo 2012, dalle 10 alle 13 e 30 marzo 2012, dalle 10 alle 13,30).
Programma dell'incontro
Diogo Gomes
(Universidade Tecnica de Lisboa):
Discrete state
mean-field games
(13 marzo 2012).
Abstract:
In this talk we will present an introduction to mean field games, a
new research area of study introduced by Caines and his co-workers,
and, independently, by Lions and Lasry. Mean field
games model situations of competition between large number of rational
agents that play
non-cooperative dynamic games under certain symmetry assumptions.
We start by an introduction to mean-field models for
discrete state problems. We will discuss
existence, uniqueness as well as applications of contraction and
monotonicity properties. Then we will consider N+1 player symmetric games and
consider the limit problem as the number of players converges to
infinity. The last part of the talk (if time allows) will be
dedicated to potential
mean-field games, as
well as various extensions and results for continuous state space
problems.
Maria Soledad Aronna
(Università di Padova):
Bang-singular optimal control: second order conditions and a shoooting algorithm
(9 marzo 2012).
Abstract
In this work we study optimal control problems for systems affine in the control variable. We consider bound constraints on the control, and finitely many equality and inequality constraints on the initial and final state values. First, we obtain second order necessary optimality conditions. Secondly, we derive a second order sufficient condition for the scalar control case.
Afterwards, we propose a shooting algorithm and we give a sufficient condition for its convergence. We show a link between this sufficient condition and the one obtained above. In some problems, we are able to analyse the stability of the optimal solution under small perturbations and the inversibility of the Jacobian matrix of the shooting function associated to the perturbed problem.
(joint work with J.F. Bonnans, A.V. Dmitruk, P.A. Lotito and P. Martinon)
Giovanni Colombo
(Università di Padova):
Brownian motion and exposed solutions of differential inclusions
(6 marzo 2012).
Abstract
We present a research program designed by A. Bressan and some partial results related to it.
First, we construct a probability measure supported on the space of solutions to a planar differential inclusion,
where the right hand side is a Lipschitz continuous segment. Such measure assigns probability one to
solutions having derivatives a.e.~equal to one of the endpoints of the segment.
Second, for a class of planar differential inclusions with Holder continuous right hand side F,
we prove existence of solutions whose derivatives are exposed points of F. Finally,
we complete the research program if the right
hand side of the differential inclusion does not depend on the state.
The proofs rely on basic properties of the scalar Brownian motion.
Joint work with V. Goncharov, with MANY thanks to P. Dai Pra.
Giuseppe M. Coclite
(Università di Bari):
An optimal harvesting problem with measure valued solutions.
(1 marzo 2012).
Abstract
In this lecture we consider a model for the harvesting of marine resources, described by an elliptic equation. Since the cost functionals have sublinear growth with respect to the pointwise intensity of fishing effort, optimal solutions are in general measure-valued. For the control problem, we prove the existence of optimal strategies.
The results were obtained in collaboration with Professors Alberto Bressan and Wen Shen.
Piero Marcati
(Università dell'Aquila):
Nonlinear Schrodinger equations and quantum fluids with critical nonlinear
and Hartree potentials.
(28 febbraio 2012).
Abstract
I shall introduce to important tools in the modern theory of Nolinear
Schrodinger equation and then I will outline some recent results (in
collaboration with P.Antonelli and G.Staffilani) regarding scattering and
asymptotic completness in presence of nonlinear and Hartree potentials in
the critical case.
I will explain the basic ingredients to get estimetes (e.g. Morawetz
estimates, Tao estimates, etc) then I will underline math difficulties due
to the Hartree potentials at critical scales and their interaction with
the study of quantum fluids.
Olivier Gueant
(Université Paris-Diderot):
Games with infinitely many players: the mean field games approach.
(13 febbraio 2012).
Abstract: The PDEs corresponding to mean field games are two strongly
coupled PDEs: one (backward) Hamilton-Jacobi-Bellman equation and one
(forward) transport equation. We will present these equations and the
intuition behind them and prove a very general criterion for
uniqueness. Existence will be tackled in the case of quadratic
hamiltonians for which many results are known and will be proved
(constructive schemes, comparison principle,...).
Andrea Calogero
(Università di Milano Bicocca):
The Hopf-Lax formula in Carnot groups: an approach with optimal control theory.
(1 febbraio 2012).
Abstract
Peter R. Wolenski
(Louisiana State University):
Hamilton-Jacobi theory for stratified systems and for problems
with state constraints and reflected dynamics
(25 gennaio 2012).
Abstract:
Bressan and Hong recently introduced an optimal control model (called a stratified system) whose dynamics have structured discontinuities in the state variable. Their main result was a viscosity-type characterization of the value function which required different Hamiltonian functions in the definitions of lower and upper solution. We shall discuss this model in detail, and by inserting additional assumptions on the data, present a characterization of the value function as the lsc solution to a bilateral Hamilton-Jacobi equation (utilizing only one Hamiltonian function). Our methods also provide a new approach to state constrained problems and to problems with reflected dynamics, in which we show how so-called boundary trajectories can be analyzed directly rather than require approximation by tracking tools.
Cyrill Muratov
(New Jersey Institute of Technology):
Gamma-convergence for pattern forming systems with competing interactions.
(18 gennaio 2012).
Abstract: I will discuss a problem of energy-driven pattern formation, in which the appearance of two distinct phases caused by short-range attractive forces is frustrated by a long-range repulsive force. I will focus on the regime of strong compositional asymmetry, in which one of the phases has very small volume fraction, thus creating small droplets of the minority phase in a sea of the majority phase. I will present a setting for the study of Gamma-convergence of the governing energy functional in the regime leading to many droplets. The Gamma-limit and the properties of almost minimizers with prescribed limit density will then be established in the important physical case when the long-range repulsive force is Coulombic in two space dimensions.
This is joint work with D. Goldman and S. Serfaty.
Lisa Beck
(Hausdorff Center For Mathematics, Bonn):
(Non-)uniqueness of graphs of least gradient - theory and examples .
(2 dicembre 2011).
Abstract: In this talk we will address some characteristics of minimization
problems concerning variational integrals of linear growth. As a model case we
study the minimization of the integral
$$\int_{\Omega} \sqrt{1 + |Dw|^2} dx$$
in Dirichlet classes of (vector-valued) function w. We first give some
heuristics on existence and uniqueness (up to additive constants) of generalized
minimizers. We then provide several classical examples (from the scalar case,
i.e. from the non-parametric least area problem) and we investigate in more
detail the phenomenon of non-uniqueness, which is closely related to the
possible non-attainment of the boundary values.
Dario Monticelli (Università di Milano):
On some fully nonlinear equations with invariances on the Heisenberg group.
(29 novembre 2011).
Abstract:
In this talk we will provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the setting of CR geometry of the result proved in the Euclidean setting by A. Li and Y.Y. Li (2003). We will also show a comparison principle for solutions of second order fully nonlinear CR invariant equations
defined on bounded domains of the Heisenberg group and a comparison principle for solutions of a family of second order fully nonlinear equations on a punctured ball.
Stefania Patrizi (Universidade Tecnica di Lisbona):
Stochastic homogenization for porous medium type equations
(23 novembre 2011).
Abstract:
In a recent paper, Ambrosio, Frid and Silva study an homogenization problem for a porous medium type equation with a stationary continuous process as a stiff oscillatory external source, on a compact probability space. We extend their result to the general stochastic setting, removing the compactness assumption. Our approach is based on the notion of kinetic solution for parabolic conservation laws and the theory of stochastic two-scale convergence.
Puduru Viswanadha Reddy (Tilburg University, Netherlands):
Optimal Management and Differential Games in the Presence of Threshold Effects - The Shallow Lake Model
(2 novembre 2011).
Abstract:
We study optimal control and differential games in a pollution control model called the shallow lake problem. The production function of this model is nonlinear (convex-concave) and as a result, the optimal vector field is known to display several interesting qualitative behaviors such as existence of multiple steady states and Skiba points. In this talk, we approximate this nonlinearity with simple and hysteresis switching and highlight the differences with the smooth case. We discuss some results and open issues.
Michael Goldman (Ecole Polytechnique, Parigi ):
Convexity of the minimizers of some variational problems in the Wiener space
(26 ottobre 2011).
Abstract:
"In this talk I will present a convexity result for variationnal problems in the Gauss and Wiener spaces. The proof is based on the approximation of this infinite dimensional problem by finite dimensional ones via Gamma-convergence. The convexity of the minimizers of the finite dimensional problem is obtained using the ideas of Alvarez, Lasry and Lions. A central point in the proof of the Gamma-convergence is the study of representation and relaxation of integral functionals in this setting."
