Organizers: Giovanni Catino and Fabio Cipriani

**Matteo Santacesaria**, Politecnico di Milano,

*Some inverse boundary value problems for PDEs: theory and applications*, Wednesday, May 11, 2016, time 16:15 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
In this talk we will focus on two inverse boundary value problems, the Calderón problem and the Gelfand-Calderon problem. The first concerns the reconstruction of an electrical conductivity from voltage and current measurements on the boundary of an object; its related imaging method is called Electrical Impedance Tomography and has applications from medical imaging to non destructive testing. In the Gelfand-Calderon problem one wants to reconstruct a potential in the Schrödinger equation from some information of its solutions at the boundary of a domain (Dirichlet to Neumann map). This problem can be seen as a model for acoustic tomography, namely with applications in geophysical prospecting.
We will first discuss theoretical properties of these problems, in particular their ill-posedness and stability estimates. In particular we will review some classical strategy to attack these problems, based on the so-called complex geometrical optics solutions and inverse scattering theory. Then we will present a new reconstruction method able to detect singularities of a conductivity from the Dirichlet-to-Neumann map: this is based on some microlocal properties of our PDE. Numerical results will be presented as well. The latter is an ongoing project in collaboration with A. Greenleaf, M. Lassas, S. Siltanen and G. Uhlmann.
**Sergio Cacciatori**, Università degli Studi dell'Insubria,

*Onde gravitazionali cento anni dopo*, Friday, April 22, 2016, time 11:00 o'clock, Aula seminari 3° piano

**Abstract:****Abstract:**
Vorrei fare una breve introduzione alla teoria della relatività generale di Einstein partendo dalle riflessioni di base che hanno portato Albert Einstein a rivoluzionare (tra tante altre cose) la concezione di spazio tempo ed in particolare la teoria della gravitazione universale. Dopo aver ricordato le equazioni di Einstein che legano la geometria dello spazio-tempo alla distribuzione di materia in esso contenuta, discuteremo brevemente alcune conseguenze di carattere generale per concentraci sulla loro linearizzazione ed in particolare la deduzione dell'esistenza delle onde gravitazionali, predette dal padre della relatività generale nel 1916 e di cui, per la prima volta, è saga segnalata una rivelazione diretta solo 100 anni dopo. Cercheremo di sottolineare quali siano le sfide aperte, soprattutto dal punto di vista dello studio analitico delle equazioni, per avere una comprensione completa del fenomeno che includa una definitiva conferma sperimentale.
**Bozhidar Velichkov**, Université Grenoble Alpes,

*Regularity of optimal sets for spectral functionals*, Wednesday, April 20, 2016, time 11:30 o'clock, Aula seminari 3° piano

**Abstract:****Abstract:**
We consider the variational shape optimization problem of the minimization of the sum of the first $k$ Dirichlet eigenvalues of a set $\Omega$ under a volume constraint $|\Omega|=1$. We prove that the free boundary of the optimal set is $C^{1,\alpha}$ regular up to a set of zero (d-1)-Hausdorff measure. The optimal set is a solution of a free boundary problem of Alt-Caffarelli type involving vector valued functions. We will dedicate most of our attention to the study of the local minimizers for this free boundary problem.
**Matteo Muratori**, Università degli Studi di Pavia,

*Symmetry results in Caffarelli-Kohn-Nirenberg interpolation inequalities*, Wednesday, April 13, 2016, time 16:15 o'clock, Aula seminari 6° piano

**Abstract:****Abstract:**
We investigate the structure of functions that optimize a special family of weighted interpolation inequalities of Caffarelli-Kohn-Nirenberg type (Compositio Math. 1984). The spatial dimension is greater than or equal to 3 and the weight appearing in the L^p norms is an inverse power with exponent between 0 and 2. The non-weighted case, associated with standard Gagliardo-Nirenberg inequalities, has thoroughly been investigated by M. Del Pino and J. Dolbeault in a remarkable paper (JMPA 2002), where they prove that optimal functions coincide with explicit profiles of Aubin-Talenti type. If the exponent ranges strictly between 0 and 2, suitable analogues of Aubin-Talenti-type profiles continue to exist. However, the main issue is related to radial symmetry. Indeed, as soon as optimal functions are radial, then they are necessarily of Aubin-Talenti type. Because of the weight, standard Schwarz symmetrization techniques fail: we have therefore to exploit a completely different method. First of all, by means of a concentration-compactness analysis, we prove that optimal functions converge to the Aubin-Talenti profiles as the exponent of the power of the weight tends to zero. Then we proceed by contradiction with an argument that involves angular derivatives of possibly non-radial optimal functions, which allows us to show that radial symmetry holds at least when the exponent is close to zero. This is a joint work with J. Dolbeault and B. Nazaret.
**Maria Colombo**, University of Zurich,

*Flows of non-smooth vector fields and applications to PDEs*, Wednesday, March 30, 2016, time 16:00 o'clock, Sala Consiglio 7° piano

**Abstract:****Abstract:**
The classical Cauchy-Lipschitz theorem shows existence and uniqueness of the flow of any sufficiently smooth vector field in R^d. In 1989, Di Perna and Lions proved that Sobolev regularity for vector fields, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. Their theory relies on a growth assumption on the vector field which prevents the trajectories from blowing up in finite time; in particular, it does not apply to fast-growing, smooth vector fields.
In this seminar we give an overview of the topic and we introduce a notion of maximal flow for non-smooth vector fields which allows for finite-time blow up of the trajectories. We show existence and uniqueness under only local assumptions on the vector field and we apply the result to a kinetic equation, the Vlasov-Poisson system, where we describe the solutions as transported by a suitable flow in the phase space. This allows, in turn, to prove existence of weak solutions for general initial data.
This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni - Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola
**Juan Luis Vazquez**, Universidad Autonoma de Madrid,

*Fractional Diffusion Equations. Theory on bounded domains*, Friday, March 11, 2016, time 11:00 o'clock, Sala Consiglio 7° piano

**Abstract:****Abstract:**
In this talk I will report on some of the progress made by the author and collaborators on the topic of nonlinear diffusion equations involving long distance interactions in the form of fractional Laplacian operators. In recent work with Bonforte and Sire we address the solution of Dirichlet problems in bounded domains, where the correct definition of what is the operator has a number of possible answers leading to different qualititative and quantitative results.
This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate
parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni
- Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration
number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola