Program

This Two-day Workshop on Nonlinear Analysis aims to bring together people involved in different branches of nonlinear analysis, trying to promote new scientific relations.

See below for the timetable and the abstracts (pdf version can be found here)

Timetable

Monday 29

• 15:00 - 15:35  Elvira Zappale Homogenization of supremal functionals in the vectorial case (via Lᴾ approximation)

• 15:40 - 16:15  Giuseppina Di Blasio Noncoercive anisotropic operator with a singular convection term

Coffee break

• 16:45 - 17:20  Francesca Colasuonno Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains

• 17:25 - 18:00  Francesco Della Pietra On the first Robin eigenvalue of nonlinear elliptic operators

Tuesday 30

• 09:00 - 09:35  Fabio Bagagiolo A single player and a mass of agents: A pursuit evasion-like game

• 09:40 - 10:15 Elsa Marchini The Maximum Principle for Lumped-Distributed Control Systems

Coffee break

• 10:45 - 11:20  Riccarda Rossi BV curves of measures and the continuity equation

• 11:25 - 12:00  Giuliano Lazzaroni On the wave equation in time-dependent domains and the problem of dynamic debonding

Abstracts

Fabio Bagagiolo (Università di Trento)

A single player and a mass of agents: A pursuit evasion-like game

Abstract: We study a finite-horizon differential game of pursuit-evasion like, between a single player and a mass of agents. The player and the mass directly control their own evolution, which for the mass is given by a first order PDE of transport equation type. Using also an adapted concept of non-anticipating strategies, we derive an infinite dimensional Isaacs equation, and by dynamic programming techniques we prove that the value function is the unique viscosity solution on a suitable invariant subset of a Hilbert space.

Francesca Colasuonno (Università di Bologna)

Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains

Abstract: In this talk, I will present an existence result for the Dirichlet problem associated with the elliptic equation  

- 𝚫𝑢 + 𝑢 = 𝑎(𝑥)|𝑢|ᵖ-²𝑢

set in an annulus or an exterior domain of Rᴺ, N ≥ 3. Here p>2 is allowed to be supercritical in the sense of Sobolev embeddings, and 𝑎 is a positive weight with additional symmetry and monotonicity properties, which are partially shared by the exhibited solution.

In the special case of radial weight 𝑎, such an existence result ensures the multiplicity of nonradial solutions. The proofs rely on variational techniques in invariant convex cones. 

This is joint work with Alberto Boscaggin (Università di Torino), Benedetta Noris (Politecnico di Milano), and Tobias Weth (Goethe University of Frankfurt).

Francesco Della Pietra (Università Federico II di Napoli)

On the first Robin eigenvalue of nonlinear elliptic operators

Abstract: In this talk, I will discuss some optimal upper and lower bounds for the first Robin eigenvalue of a nonlinear elliptic operator, with Robin boundary conditions. These bounds are expressed in terms of geometrical quantities related to the domain. If I will have time, I will also discuss the case where p approaches 1.

Giuseppina Di Blasio (Università degli Studi della Campania)

Noncoercive anisotropic operator with a singular convection term

Abstract: The aim of this seminar is to outline some results obtained in recent years concerning a class of anisotropic elliptic equations with the coefficients of a convection term belonging to some suitable Marcinkiewicz spaces. This talk is based upon a joint project in collaboration with F. Feo and G. Zecca.

Giuliano Lazzaroni (Università degli Studi di Firenze)

On the wave equation in time-dependent domains and the problem of dynamic debonding

Abstract: In models for dynamic debonding, the wave equation is set on a time-dependent domain and is coupled with a Griffith criterion for the evolution of such domain. This problem can be seen as a simplified version of dynamic fracture, at least in dimension one, where solutions can be determined in closed form. In dimension two the problem is more complex due to the shape of the debonding front, which may affect wave propagation. In the talk I will present some abstract results for the wave equation in time-dependent domains, bringing to a definition of dynamic energy release rate and to a formulation of the coupled problem in a general setting. I will also show how such problem can be solved assuming that solutions are radial. From joint works with R. Molinarolo, F. Riva, F. Solombrino.

Elsa Marchini (Politecnico di Milano)

The Maximum Principle for Lumped-Distributed Control Systems

Abstract: Classical optimal control concerned control strategies for dynamical systems modelled as a control differential equation, state trajectories for which evolve in a finite dimension space. Examples of application were in aerospace, chemical engineering and mathematical economics. A later development was optimal control of infinite dimensional systems covering for example wave or heat equations, characterized by an infinite dimensional state space. 

In this talk we report on a work concerning the optimal control of a subclass of systems having an infinite dimensional state space, called lumped-distributed systems. Such a system is an interconnection of distributed and finite dimensional (lumped) systems, arising in robotics (control of masses connected by flexible rods). This description also arises in communication systems where, for example, a transmission line has an active load, in thermal systems where a distributed thermal channel interacts with heat sinks and sources, and in optimal control of hereditary systems, when the dynamic constraint is reformulated as a `delay-free' evolution equation with an infinite dimensional state space.

A number of difficulties are encountered when we attempt to model the derivation of necessary conditions for problems with infinite dimensional state spaces on the standard, finite dimensional theory. The most notable of these is that, for problems with constraints and problems with non-smooth data, perturbational methods, based on approximation and passage to the limit, may fail. These difficulties are all, broadly speaking, connected with the fact that general infinite dimensional systems lack the compactness properties that are a key element in finite dimensional analysis.

This work identifies a class of infinite dimensional systems, encountered across a range of engineering disciplines,  where the difficulties discussed above can be overcome. 

Joint work with Richard Vinter.

Riccarda Rossi (Università degli Studi di Brescia)

BV curves of measures and the continuity equation

Abstract: Representation results for Lipschitz (or even absolutely continuous) curves 𝜇 : [0,T] ⟶ 𝒫ᵖ(Rᵈ), p>1, with values in the Wasserstein space (𝒫ᵖ(Rᵈ),Wᵖ) of Borel probability measures in Rᵈ with finite p-moment provide a crucial tool to study evolutionary PDEs and geometric problems in a measure-theoretic setting.

They are strictly related to corresponding representation results for measure-valued solutions

to the continuity equation, as a superposition of absolutely continuous curves solving a suitable differential equation.

In this talk we discuss the validity and the appropriate formulation of the above results in the case p=1, for the space of probability measures with finite moment 𝒫¹(Rᵈ) endowed with the metric W¹. We will thus provide a suitable version of the superposition principle for curves of measures in 𝒫¹(Rᵈ) that are only of bounded variation with respect to the time variable.

Joint work with Stefano Almi (Napoli) and Giuseppe Savaré (Milano). 

Elvira Zappale (Sapienza Università di Roma)

Homogenization of supremal functionals in the vectorial case (via Lᴾ approximation)

Abstract: The aim of this talk consists of presenting a homogenized supremal functional rigorously derived via power-law approximation by functionals of the type ess sup f(𝑥/𝜖, D𝑢), when Ω is a bounded open set of Rⁿ and 𝑢 ∈ W¹,∞(Ω;Rᵈ). The homogenized functional is also deduced directly in the case where the sublevel sets of f(𝑥,⋅) satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals in the vectorial setting. 

This is a joint work with Lorenza D'Elia (TU Wien) and Michela Eleuteri (Università di Modena e Reggio Emilia).