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.

ENRICO VITALI, Universita` di Pavia, Un modello variazionale discreto del secondo ordine, Wednesday, March 16, 2011, time 11:00 o'clock, Aula Seminari III piano
Abstract:Abstract:
We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions with bounded hessian, and we give an upper and a lower bound in terms of the Blake and Zisserman energy.

GIACOMO CARBONI, Universita` del Salento, Image segmentation e inpainting: approccio variazionale e modellizzazione numerica, Wednesday, March 16, 2011, time 12:00 o'clock, Aula Seminari III piano
Abstract:Abstract:
Si presentano risultati noti sull esistenza di minimi per il funzionale di Blake & Zisserman e l applicazione dello stesso al problema della segmentazione di immagini. Si introduce un metodo di approssimazione numerica per le equazioni di Eulero-Lagrange associate e si presentano i risultati dell analisi. Successivamente si presentano i risultati di esistenza per i minimi di un nuovo funzionale del secondo ordine e la sua applicazione al problema dell inpainting. La ricerca di una soluzione per le equazioni di Eulero-Lagrange associate viene affrontata in maniera analoga a quanto fatto nel caso della segmentazione. A completare la presentazione, alcuni accenni all analisi del metodo ed esempi.

Helmut Abels, Universitaet Regensburg, On a new diffuse interface model for incompressible two-phase flows with different densities, Wednesday, March 02, 2011, time 16:15 o'clock, Aula Seminari VI piano
Abstract:Abstract:
We discuss different models for a two-phase flow of two immiscible, incompressible fluids in the case when the densities of the fluids are different. In particular we will present a new thermodynamically consistent diffuse interface model and compare it with the known models. Such models were introduced to describe the flow when singularities in the interface, which separates the fluids, (droplet formation/coalescence) occur. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed. We will briefly discuss its derivation and its sharp interface limits. Moreover, we present a recent result on existence of weak solutions for this model.

Aris Daniilidis, Universita Autonoma Barcelona, Behavior of the gradient flow: the convex case, Wednesday, Febraury 16, 2011, time 15:00 o'clock, Aula Seminari VI piano
Abstract:Abstract:
The classical Lojasiewicz inequality and its extension to o-minimal structures by K. Kurdyka has a considerable impact on the analysis of gradient-like methods and related problems. In this talk we shall discuss alternative characterizations of this type of inequality via the notion of a defragmented gradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka-Lojasiewicz inequality is satisfied. Another characterization in terms of talweg lines will be given. In the convex case these results are significantly reinforced, allowing in particular to establish a kind of asymptotic equivalence for discrete gradient methods and continuous gradient curves.