Organizers: Giovanni Catino and Fabio Cipriani

**Guglielmo Feltrin**, Politecnico di Torino,

*Positive solutions to indefinite problems: a topological approach*, Thursday, December 06, 2018, time 15:30, Aula seminari 3° piano

**Abstract:****Abstract:**
In this seminar, we present some recent existence and multiplicity results for positive solutions of boundary value problems associated with second-order nonlinear indefinite differential equations. More precisely, we deal with the ordinary differential equation
u?? + a(t)g(u) = 0,
where a: [0,T] ? R is a Lebesgue integrable sign-changing weight and g: [0,+?[ ? [0,+?[ is a continuous nonlinearity.
We focus on the periodic boundary value problem and on functions g(u) with superlinear growth at zero and at infinity (including the classical superlinear case g(u) = up, with p > 1). Exploiting a new approach based on topological degree theory, we show that there exist 2m ? 1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large. In this manner, we give a complete answer to a question raised by Butler (JDE, 1976) and we solve a conjecture by G ?omez-Ren ?asco and L ?opez-G ?omez (JDE, 2000). The method also applies to Neumann and Dirichlet boundary conditions and, furthermore, provides a topological approach to detect infinitely many subharmonic solutions and globally defined positive solutions with chaotic behaviour.
Thereafter, we illustrate other directions for the research on indefinite problems: super-sublinear problems, models in population genetics, and also problems involving more general differential oper- ators, as the Minkowski-curvature one or the one-dimensional p-Laplacian. Exact multiplicity results and indefinite problems in the PDE setting are also discussed.
The talk is based on joint works with Alberto Boscaggin (University of Torino), Elisa Sovrano (University of Porto) and Fabio Zanolin (University of Udine) and on the book “Positive Solutions to Indefinite Problems. A Topological Approach” (Frontiers in Mathematics, Birkh ?auser/Springer, 2018).
**Luigi Vezzoni**, Università degli Studi di Torino,

*The Quantitative Alexandrov Theorem in Space forms*, Tuesday, November 27, 2018, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
The talk focuses on a recent generalization of a classical result of Alexandrov. The celebrated Alexandrov's Soap Bubble Theorem states that the spheres are the only closed (i.e. compact and without boundary) constant mean curvature hypersurfaces embedded in the Euclidean space. The theorem has been generalized to the hyperbolic space and to the hemisphere and to a large class of curvature operators. The main result of the talk is a quantitative version of Alexandrov's theorem which I've obtained in collaboration with Giulio Ciraolo and Alberto Roncoroni by using a quantitative study of the method of the moving planes. The theorem implies a new pinching Theorem for hypersurfaces in space forms.
**Alessandro Zilio**, Université Paris Diderot,

*Predator-prey model with competition, the emergence of territoriality*, Tuesday, October 30, 2018, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
I will present a series of works in collaboration with Henri Berestycki (PSL), dealing with systems of predators interacting with a single prey. The system is linked to the Lotka-Volterra model of interaction with diffusion, but unlike more classic works, we are interested in studying the case where competition between predators is very strong: in this context, the original domain is partitioned in different sub-territories occupied by different predators. The question that we ask is under which conditions the predators segregate in packs and whether there is a benefit to the hostility between the packs. More specifically, we study the stationary states of the system, the stability of the solutions and the bifurcation diagram, and the asymptotic properties of the system when the intensity of the competition becomes infinite.
**Shmuel Zamir**, The Hebrew University,

*Strategic Use of Seller Information in Private-Value First-Price Auctions*, Monday, October 22, 2018, time 11:00, Politecnico di Milano, Dipartimento di Matematica, Sala del Consiglio 7° piano

**Abstract:****Abstract:**
In the framework of a private-value-first-price auction, we consider the seller as a player in a game with the buyers in which he has private information about their realized valuations. We ask whether the seller can benefit by using his private information strategically. We find that in fact, depending upon his information, set of signals, and commitment power the seller may indeed increase his revenue by strategic transmission of his information. For example, in the case of two buyers with values distributed independently and uniformly on [0,1], a seller informed of the private values of the buyers, can achieve a revenue close to 1/2 by sending verifiable messages (compared to 1/3 in the standard auction), and this is the largest revenue that can be obtained with any signalling strategy.
**Lorenzo Toniazzi**, University of Warwick,

*Caputo Evolution Equations with time-nonlocal initial condition*, Tuesday, October 09, 2018, time 15:15, Aula Seminari 3° piano

**Abstract:****Abstract:**
Consider the Caputo evolution equation (EE) $\partial_t^\beta u =\Delta u$ with initial condition $\phi$ on $\{0\}\times\mathbb R^d$, $\beta\in(0,1)$. As it is well known, the solution reads $u(t,x)=\mathbf E_x[\phi(B_{E_t})]$. Here $B_t$ is a Brownian motion and the independent time-change $E_t$ is an inverse $\beta$-stable subordinator. The fractional kinetic $B_{E_t}$ is a popular model for subdiffusion \cite{Meerschaert2012}, with remarkable universality properties \cite{BC11,Hai18}.\\
We substitute the Caputo fractional derivative $\partial_t^\beta$ with the Marchaud derivative. This results in a natural extension of the Caputo EE featuring a \emph{time-nonlocal initial condition} $\phi$ on $(-\infty,0]\times\mathbb R^d$. We derive the new stochastic representation for the solution, namely $u(t,x)=\mathbf E_x[\phi(-W_t,B_{E_t})]$. This stochastic representation has a pleasing interpretation due to the non-obvious presence of $W_t$, elucidating the notion of time-nonlocal initial conditions. Here $W_t$ denotes the waiting/trapping time of the fractional kinetic $B_{E_t}$. We discuss classical-wellposedness \cite{T18}, and time permitting weak-wellposedness \cite{DYZ17,DTZ18} for the respective extensions of Caputo-type EEs (such as in \cite{chen,HKT17}).
Bibliography:
Barlow, \u Cern\'y (2011). Probability theory and related fields, 149.3-4: 639-673.
Chen, Kim, Kumagai, Wang (2017). arXiv:1708.05863.
Du, Toniazzi, Zhou (2018). Preprint. Submitted in Sept. 2018.
Du, Yang, Zhou (2017). Discrete and continuous dynamical systems series B, Vol 22, n. 2.
Hairer, Iyer, Koralov, Novikov, Pajor-Gyulai (2018). The Annals of Probability, 46(2), 897-955.
Hern\'andez-Hern\'andez, Kolokoltsov, Toniazzi (2017). Chaos, Solitons \& Fractals, 102, 184-196.
Meerschaert, Sikorskii (2012). De Gruyter Studies in Mathematics, Book 43.
Toniazzi (2018). To appear in: Journal of Mathematical Analysis and Applications. arXiv:1805.02464.
**Luca Ratti**, Politecnico di Milano,

*An inverse boundary value problem arising from cardiac electrophisiology*, Tuesday, September 18, 2018, time 15:15, Politecnico di Milano, Dipartimento di Matematica, Aula Seminari 3° Piano

**Abstract:****Abstract:**
The cardiac electrical activity can be comprehensively described throughout the monodomain model, consisting of a semilinear parabolic equation coupled with a nonlinear ordinary differential equation.
In my talk, I will introduce the inverse problem of identifying conductivity inhomogeneities in the monodomain system, taking advantage of data acquired on the boundary of the domain. Due to the complexity of the task, I will first tackle the stationary counterpart of the problem, regarding which it is possible to formulate well-posedness results both for the forward and for the inverse problem, and to rigorously introduce reconstruction procedures. Similar results are then generalized to the full complexity of the original model.
Throughout the presentation, I will focus on the problem of localizing small size inhomogeneities, as well as arbitrarily large ones, by means of the constraint optimization of a suitable misfit functional. The first task is achieved by relying on an asymptotic expansion of the boundary voltage with respect to the size of the inclusion, and employing tools from the topological optimization theory. The second issue is analyzed by means of the regularization theory of inverse problems and introducing a convenient relaxation of the optimization problem. The theoretical results are supported by numerical experiments, which are exhaustively reported.
This is a joint work with Elena Beretta, Cristina Cerutti, Cecilia Cavaterra, Andrea Manzoni and Marco Verani.