Federica Sani, Università degli Studi di Milano, Trudinger-Moser and Adams inequalities with the exact growth condition, Friday, May 20, 2016, time 11:00 o'clock, Aula seminari 6° piano
Abstract:Abstract:
The Trudinger-Moser inequality is a substitute for the well known Sobolev embedding Theorem when the limiting case is considered. Adams' inequality is the complete generalization of the Trudinger-Moser inequality to the case of Sobolev spaces involving higher order derivatives. In this talk, we discuss the optimal growth rate of the exponential-type function in Trudinger-Moser and Adams inequalities when the problem is considered in the whole space R^n.

Benedetta Pellacci, Università di Napoli , Nonlinear Problems in Exterior Domains, Wednesday, May 18, 2016, time 16:15 o'clock, Aula seminari 3° piano
Abstract:Abstract:
We will study the existence of a positive solution for non-linear problems in exterior domains. This topic is related to the search of non-symmetric solutions of saturable or non-linear Schrödinger equations when the domain is not symmetric. This a joint work with Liliane Maia (University of Brasilia).

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.