Organizers: Giovanni Catino and Fabio Cipriani

**Silvia Villa**, Politecnico di Milano,

*Convergence of proximal gradient methods*, Wednesday, November 16, 2016, time 13:15 o'clock, Sala del consiglio 7° piano

**Abstract:****Abstract:**
First order methods have recently been widely applied to solve convex optimization problems in a
variety of areas including machine learning and signal processing.
In particular, proximal gradient algorithms (a.k.a. forward-backward splitting algorithms) and their
accelerated variants have received considerable attention. These algorithms are easy to implement
and suitable for solving high dimensional problems thanks to the low memory requirement of each iteration.
In this talk I will present some recent convergence results for this class of methods.
**Giovanni Alberti**, ETH, Zurigo,

*Absence of critical points of solutions to the Helmholtz equation in 3D*, Wednesday, October 12, 2016, time 16:15 o'clock, Aula seminari 3° piano

**Abstract:****Abstract:**
In this talk I will discuss a method to prove the absence of critical points for the Helmholtz equation in 3D. The key element of the approach is the use of multiple frequencies in a fixed range, and the proof is based on the spectral analysis of the associated problem. This question is strictly connected with the Rado-Kneser-Choquet theorem, whose direct extension to the Helmholtz equation or to three dimensions is not possible.
**Edi Rosset**, Università di Trieste,

*Global stability for an inverse problem in soil-structure interaction*, Wednesday, July 13, 2016, time 16:15 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
An important issue in structural building design is the soil–structure interaction. We make reference to the Winkler model and consider the inverse problem of determining the Winkler subgrade reaction coefficient k(x) of a slab foundation modelled as a thin elastic plate clamped at the boundary and loaded by a concentrated force. We investigate whether the coefficient k(x) depends continuously on the transversal deflection taken at the interior points. We prove a global Hölder stability estimate under some reasonable regularity assumptions on the unknown Winkler coefficient.
**Maurizio Garrione**, Università degli Studi di Milano-Bicocca,

*Multiple Neumann solutions of some second order ODEs with indefinite weight*, Wednesday, June 08, 2016, time 16:15 o'clock, Aula seminari 3° piano

**Abstract:****Abstract:**
We consider the Neumann problem associated with the scalar second order ODE u''+q(t)g(u)=0, where q is a sign-changing weight and g is a positive function satisfying sublinear growth assumptions. Through the use of a suitable change of variables, the problem is led back to the study of Neumann solutions for a forced perturbation of an autonomous planar system, which we study by means of a shooting technique. We illustrate results of existence and multiplicity of solutions depending in a quite precise way on mean properties of the weight.
**Scott Rodney**, Cape Breton University, Canada,

*A Meyers-Serrin Theorem for Degenerate Sobolev Spaces with an application to degenerate $p$-Laplacians*, Wednesday, June 01, 2016, time 16:15 o'clock, Aula seminari 3° piano

**Abstract:****Abstract:**
It is well understood that degenerate elliptic PDEs in divergence form play an important role in many areas of mathematics. For a non-negative definite measurable matrix valued function $A(x)$ and $1?p<\infty$, the degenerate matrix-weighted Sobolev spaces $H^{1,p}_A(\Omega)$ (defined as a closure of $C^\infty(\Omega)$) and $W^{1,p}_A(\Omega)$ (defined as a collection of functions with locally integrable distributional derivatives) play a central role in regularity theory and applications. In this talk, I present joint work with D. Cruz-Uribe and K. Moen that gives a sharp condition on the matrix function A for the equality $H^{1,p}_A(\Omega) = W^{1,p}_A(\Omega)$.
**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.