Organizers: Giovanni Catino and Fabio Cipriani

**Shuai Lu**, Fudan University, Shanghai, China,

*On the Inverse Problems for the Coupled Continuum Pipe Flow model for flows in karst aquifers*, Tuesday, December 06, 2011, time 11:30, Aula seminari III piano

**Abstract:****Abstract:**
We investigate two inverse problems for the coupled continuum pipe flow (CCPF) model which describes the fluid flows in karst aquifers. After generalizing the well-posedness of the forward problem to the anisotropic exchange rate case which is a
space-dependent variable, we present the uniqueness of this parameter by measuring the Cauchy data. Besides, the uniqueness of the geometry of the conduit by the Cauchy data is verified as well. These results enhance the practicality of the CCPF model.
**Eduardo Teixeira**, Universidade Federal do Ceara, Brazil,

*Nonvariational singular elliptic equations and their geometries*, Friday, November 18, 2011, time 14:00 o'clock, Aula seminari VI piano

**Abstract:****Abstract:**
In this talk I will discuss fine geometric properties of
solutions to fully nonlinear elliptic equations with singular
potentials. Singularities occurs along the (unkown) free surface {u=0}
and the degree of the singularity determines optimal regularity
estimates for u though the singular level set. These classes of
equations are related to important free boundary problems that were
previously accessed only through variational considerations.
**Youcef Mammeri**, Université de Picardie Jules Verne - Amiens,

*Asymptotic behavior of small solutions of the Benjamin-Ono equations*, Friday, November 18, 2011, time 15:00 o'clock, Aula Seminari VI piano

**Abstract:****Abstract:**
We study the behavior of small solutions depending on time of the generalized and regularized Benjamin-Ono equation in both continuous and periodic contexts. In particular, we prove that these solutions remain small. Moreover, we show that solutions are decreasing in the continuous case.
**Cristina Trombetti**, Universita di Napoli Federico II,

*SU UN PROBLEMA DI BISEZIONE NEL PIANO E DISUGUAGLIANZE DI SOBOLEV- POINCARE*, Thursday, May 26, 2011, time 12:15 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
We prove a long standing conjecture concerning the fencing problem in the plane: among planar convex sets of given area, prove that the disc, and only the disc maximizes the length of the shortest area-bisecting curve. Although it may look intuitive, the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so-called Auerbach triangle.
**Patrick Guidotti**, University of California, Irvine,

*A New Forward-Backward Regularization of the Perona-Malik equation*, Tuesday, March 22, 2011, time 15:00 o'clock, Aula Seminari VI piano

**Abstract:****Abstract:**
The Perona-Malik equation (PME) is a forward-backward nonlinear diffusion which was proposed
in the context of image processing as an image enhancement tool capable of preserving sharp
features such as edges. To this day its mathematical nature has not been fully understood in spite of many an attempt. After a brief historical overview of the mathematical results available for the equation and its many regularizations/relaxations, the talk will introduce and analyze a novel,
rather natural, regularization which will shed light on the nature of PME. The regularization is
quite mild in that PME is regularized by a family of forward-backward equations, the solutions of
which are, however, better behaved.