Organizers: Giovanni Catino and Fabio Cipriani

**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.
**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.