Organizers: Giovanni Catino and Fabio Cipriani

**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.
**Francesco Tulone**, Università di Palermo,

*HK-type integral and application*, Tuesday, Febraury 17, 2015, time 11:00 o'clock, Aula seminari MOX, VI piano

**Abstract:****Abstract:**
Durante il seminario si enunceranno le principali proprietà di integrali non-assolutamente convergenti e le loro applicazioni nella ricostruzione dei coefficienti di Fourier per sistemi di Walh e di Vilenkin
**Marisa Toschi**, Instituto de Matematica Aplicada del Litoral, Departamento de Matematica, Santa Fe, Argentina ,

*Elliptic operators and Muckenhoupt weights*, Thursday, December 04, 2014, time 14:30 o'clock, Aula Seminari III piano

**Abstract:****Abstract:**
Under different conditions on a domain $\Omega$, we prove some weighted a priori estimates for solutions to the Dirichlet problem when the weight is in the Muckenhoupt class $A_p(R^n)$. In particular, we are interested in the case of weights which are powers of the distance to the boundary of $\Omega$ and we study for which powers these weights belongs to $A_p(R^n)$.
These results can be extended to a metric measure space $(X;d;\mu)$
satisfying the so called Ahlfors condition, which is a particular case of space of homogeneous type. We also obtain a new family of weights in the class $A_p(X;d;\mu)$ for a general metric measure space.
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
**DEBDIP GANGULY**, POLITECNICO DI TORINO,

*PARTIAL DIFFERENTIAL EQUATIONS ON HYPERBOLIC SPACE *, Friday, November 28, 2014, time 14:30 o'clock, Aula Seminari III piano

**Abstract:****Abstract:**
In this talk, semilinear elliptic partial differential equations(PDEs) on hy-perbolic space and related problems will be presented. Several geometric problems lead to the study of the equation:
$$- \Delta _{B ^N} u - \lambda u = |u| ^ {p-2} u , u \in H ^ 1 (B ^N)$$
where $\lambda$ is a real parameter and $H^1(B^N)$ denotes the Sobolev space on the conformal ball model of the hyperbolic space. Some existence, non existence and qualitative properties of solutions of above equation will be pointed out.