Organizers: Giovanni Catino and Fabio Cipriani

**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.
**Zhe Zhou**, Chinese Academy of Sciences, Beijing,

*Rotation number of the linear Schrödinger equation with discontinuous almost periodic potentials*, Thursday, September 13, 2018, time 15:00, Aula Seminari 3° piano

**Abstract:****Abstract:**
In this talk, based on the celebrated paper [R. Johnson and J. Moser, Comm. Math. Phys., 1982], we will recover the rotation numbers of the Schrödinger equation. The essential elements in the proof are positive homogeneity and almost periodicity. From this point of view, the concept of rotation numbers may be introduced in the case of discontinuous potentials. Moreover, we will show the optimal estimate of rotation numbers in such case.
**Maurizio Garrione**, Politecnico di Milano,

*Linear and nonlinear equations for beams and degenerate plates with double piers*, Tuesday, June 19, 2018, time 15:45, Aula seminari 6° piano

**Abstract:****Abstract:**
Motivated by the phenomena observed on the occasion of the famous Tacoma Narrows Bridge collapse in 1940, we deal with some nonlinear fourth-order differential equations related to the analysis of the dynamics of suspension bridges. Following a "structural" approach, we discuss the role of the position of intermediate piers in the stability of a hinged beam, making a comparison between different notions of stability. The analysis is carried out analytically, with some help from numerics. (Joint work with Filippo Gazzola)
**Berardino Sciunzi**, Università della Calabria,

*On the Hopf boundary lemma for quasilinear problems involving singular nonlinearities and applications*, Wednesday, May 23, 2018, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
We consider positive solutions to quasilinear elliptic problems with singular nonlinearities. We provide a Hopf type boundary lemma via a suitable scaling argument that allows to deal with the lack of regularity of the solutions up to the boundary. Symmetry and monotonicity properties of the solutions follows as an application.
**Alberto Boscaggin**, Università di Torino,

*Periodic solutions to perturbed Kepler problems*, Tuesday, May 22, 2018, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
As well known (by third Kepler’s law) the Kepler problem has many periodic solutions with minimal period T (for any given T > 0). We will try to understand how many of them survive after a T-periodic external perturbation preserving the Newtonian structure of the equation. In doing this, we will be naturally led to the concept of generalized solution and to the theory of regularization of collisions in Celestial Mechanics. Joint work with Rafael Ortega (Granada) and Lei Zhao (Augsburg).