Organizers: Giovanni Catino and Fabio Cipriani

**Dario D. Monticelli**, Politecnico di Milano,

*On the Dirichlet problem of mixed type for lower hybrid waves in axisymmetric cold plasmas*, Friday, November 13, 2015, time 11:00 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
In this talk we will describe some recent results concerning the Dirichlet problem for a class of second order differential equations of mixed elliptic-hyperbolic type on suitable bounded domains of R^2, which is used as a model to describe possible heating in axisymmetric cold plasmas subjected to high frequency electromagnetic waves near certain frequencies. The presence of hyperbolicity makes the problem overdetermined for classical solutions. We will show that the problem is well-posed for weak solutions belonging to a weighted version of the classical Sobolev space H^1_0, when the datum is chosen in a suitable weighted L^2 space. We will also provide a complete spectral theory for the Dirichlet problem in the setting of weighted Lebesgue and Sobolev spaces, with some applications to semilinear equations and to equations with lower order terms. Finally we will give a variational characterisation of weak solutions, which are shown to be saddle points of an associated strongly indefinite functional.
These results are joint work with D. Lupo (Politecnico di Milano) and K.R. Payne (Università degli Studi di Milano).
**Davide Buoso**, Politecnico di Torino,

*Shape sensitivity analysis for vibrating plates*, Friday, October 30, 2015, time 11:00 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
In this talk we will consider the eigenvalues of the biharmonic operator subject to various boundary
conditions, namely Dirichlet, Neumann, Navier and Steklov ones. Note that such problems arise in the theory of linear elasticity, within the so-called Kirchhoff-Love model for the vibration of a plate. We will show that the eigenvalues of the Bilaplacian are analytic with respect to the shape, and compute Hadamard-type formulas for their differentials. Then, using the Lagrange Multiplier Theorem, we are able to show that the ball is a critical domain for any eigenvalue (under any of the boundary conditions considered). In the last part of the talk we will focus on eigenvalue shape optimization results for Neumann and Steklov problems. We will provide isoperimetric inequalities in quantitative form for the fundamental tones (i.e., the first non-trivial eigenvalues), and discuss the limiting cases.
**Corrado Mascia**, Universita' di Roma "La Sapienza",

*Hyperbolic traveling fronts: the bistable equation with relaxation*, Thursday, June 25, 2015, time 14:00 o'clock, Aula Seminari III Piano

**Abstract:****Abstract:**
The main concern of the talk is to discuss a class of hyperbolic equation
in the presence of a reaction term of Allen-Cahn type, motivated by
the assumption that the alignment of the flux term with the gradient of the unknown function is not istantaneous but delayed by the presence of a relaxation time.
After a brief overview on the derivation of such class of equations
starting from some appropriate modelling assumptions, emphasis will be given to the topic of front propagation in one dimension.
Rigorous results concerning existence and stability of planar fronts will be presented, comparing it with the corresponding results for the
standard parabolic Allen--Cahn equation.
(joint collaboration with C.Lattanzio, R.G.Plaza, C.Simeoni)
**Donatella Danielli**, University of Purdue,

*Regularity results for a class of permeability problems*, Wednesday, June 24, 2015, time 11:30 o'clock, Aula consiglio VII piano

**Abstract:****Abstract:**
In this talk we will present an overview of regularity results (both for the solution and for the free boundary) in a class of problems which arises in permeability theory. We will mostly focus on the parabolic Signorini (or thin obstacle) problem, and discuss the modern approach to this classical problem, based on several families of monotonicity formulas. In particular, we will present the optimal regularity of the solution, the classification of free boundary points, the regularity of the regular set, and the structure of the singular set. These results have been obtained in joint work with N. Garofalo, A. Petrosyan, and T. To.
We will also discuss the regularity of solutions in a related model arising in problems of semi-permeable walls and of temperature control. This is joint work with T. Backing.
**Francesco Di Plinio**, Brown University Mathematics Department,

*Pointwise convergence of Fourier series and a Calderon-Zygmund decomposition for modulation invariant singular integrals*, Friday, June 19, 2015, time 14:00 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
The Calderon-Zygmund decomposition is a fundamental tool in
the analysis of singular integral operators, prime examples of which are
the Hilbert and Riesz transforms occurring, for instance, in elliptic
PDEs. Its prime purpose is the extension of L^2 boundedness results for
operators of this type to (all) other L^p spaces. The exploited
mechanism is the extra cancellation occurring when the singular integral
operator is applied to functions with mean zero.
On the other hand, pointwise convergence of Fourier series of L^p
functions is governed by L^p bounds for maximally modulated versions of
singular integrals, for which the single zero frequency of f plays no
particular role. In this talk, we describe a novel Calderon-Zygmund
decomposition adapted to the maximally modulated setting and its
applications to both pointwise convergence of Fourier series of
functions near L^1 and sharp bounds for the bilinear Hilbert transform
near the critical exponent L^{2/3}. The presentation will be
non-technical and suitable for an advanced undergraduate audience.
Partly joint work with Ciprian Demeter (IU), Christoph Thiele (Bonn),
Andrei Lerner (Bar-Ilan U, Israel) and Yumeng Ou (Brown U).
**Filippo Dell' Oro**, Institute of Mathematics of the Academy of Sciences of the Czech Republic,

*Exponential stability for thermoelastic Bresse systems with nonclassical heat conduction*, Friday, March 06, 2015, time 14:00, Aula seminari III piano

**Abstract:****Abstract:**
We provide a comprehensive stability analysis of the thermoelastic Bresse system (also known as the circular arch problem). In particular, assuming a temperature evolution of Gurtin-Pipkin type, we establish a necessary and sufficient condition for exponential stability in terms of the structural parameters of the problem. As a byproduct, a complete characterization of the longtime behavior of Bresse-type systems with Fourier, Maxwell-Cattaneo and Coleman-Gurtin thermal laws is obtained.