Alessandra Pluda

Luigi Forcella - 25th May 2016 -

"The electrostatic limit for the 3D Zakharov system"

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We consider the Zakharov system, coupling Schrödinger-like and wave equations, which models some dynamics in a plasma. This system in its physical derivation depends on a parameter αα describing the temperature of this plasma. Large value of such parameter describes a plasma which is not very hot, so is meaningful to study the limit for α towards infinity as physical observations suggest. In this talk we prove rigorous mathematical result in this direction. (Joint work with Paolo Antonelli, GSSI, L’Aquila)

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David Tewodrose -18 May 2016-

"Large-time behavior of the heat kernel on manifolds with non-negative Ricci curvature"

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The two formulae of the heat kernel on R^n and on the hyperbolic space H^n give a first example of the fact that the geometry of a space has an incidence on the heat kernel. On general manifolds, heat kernels are almost never expressed by such explicit formulae. However, under some hypothesis on the geometry of the manifold, nice upper or lower bounds of the heat kernel can be obtained. In an article on 1986, P. Li used such bounds to obtain a first result concerning the asymptotic behavior of the heat kernel on manifolds with nonnegative Ricci curvature and maximal volume growth. In 2013, Xu obtained a similar result removing the maximal volume growth hypothesis. The object of the seminar will be to present Li's and Xu's works, stressing the use by Xu of Cheeger-Colding's theory.

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Stefano Gioffrè - 11th May 2016 -

"Limits equations arising in the vacuum Einstein constraint equations using the conformal method"

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In this seminar, we deal with the Einstein problem in general relativity, treating in particular the vacuum case. Using a trick developed by Choquet-Bruhat and Lichnerowicz - the well known "conformal method" - one can write the vacuum Einstein equations as a couple of differential equations, called constraint equations: a Yamabe-like scalar equation with a nonlinearity term, and a vector equation coupled to the first. In an article written in 2011 by Gicquaud, Humbert, Dahl it is proven a theorem which states that either the constraint equations admit solutions or another differential equation, a limit one admits a non-trivial solution. Our aim is to present this problem, explain the main ideas and present an attempt to prove non-existence theorems for the limit equations based on a fine analysis of the proof made article.

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Marcello Carioni -28th April 2016 -

"Calibrations for the Mumford-Shah functional"

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The Mumford-Shah functional is one of the most studied variational approach to image processing and edge detection, proposed by Mumford and  Shah in the late 80's. In the first part of the talk we present it, giving  basic properties, known results and conjectures about that. Then, we focus on a suitable calibration technique for this functional, introduced by Alberti, Bouchitté and Dal Maso in 1999 and we show how it can be used to prove that some candidate functions are minimizers. Lastly we will deal  with the opposite question: we ask if, given a minimizer, there always exists a calibration for it. We will state some partial results in this direction and some open problems.

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Giovanni Mascellari - 12th April 2016 -

"Funzioni armoniche e  teoremi della sfera"  

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In geometria riemanniana sono noti molti teoremi cosiddetti "della sfera", i quali danno condizioni sufficienti (in genere di tipo  analitico, riguardanti disuguaglianze tra tensori di curvatura, volume, autofunzioni ed altri oggetti geometrici) perché una data varietà sia isometrica ad una sfera. In molti di tali teoremi la sfera viene  caratterizzata come oggetto in qualche senso "estremale" rispetto ad una certa grandezza; questo ricalca un comportamento ben noto in analisi: quando si dà una disuguaglianza, è spesso significativo studiare le specificità del caso in cui questa diventi effettivamente un'uguaglianza.  Durante il seminario presenterò alcuni risultati di tipo sfera per soluzioni dell'equazione di Laplace in un dominio esterno di R^n,  evidenziando anche i collegamenti con il classico teorema della sfera di  Obata, e delineerò i punti principali della dimostrazione.

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Leonard Kreutz - 6th April 2016 -

"Optimal bounds for mixtures of ferromagnetic interactions"

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We completely describe the effective surface tension of periodic mixtures of two types of ferromagnetic interactions. This problem is linked to optimal design of networks and their metric properties. This is joint work with A. Braides.

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Annalisa Massaccesi - 16th March 2016 -

"Is a nonlocal diffusion strategy convenient for biological populations in competition?"

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In this joint work with Enrico Valdinoci, we study the convenience of a nonlocal dispersal strategy in a reaction-diffusion system with a fractional Laplacian operator. We show that there are circumstances - namely, a precise condition on the distribution of the resource - under  which a nonlocal dispersal behavior is favored. In particular, we consider the linearization of a biological system that models the interaction of  two biological species, one with local and one with nonlocal dispersal, that are competing for the same resource. We give a simple, concrete  example of resources for which the equilibrium with only the local  population becomes linearly unstable. In a sense, this example shows that  nonlocal strategies can become successful even in an environment in which  purely local strategies are dominant at the beginning, provided that the resource is sufficiently sparse. Indeed, the example considered presents a high variance of the distribution of the dispersal, thus suggesting that the shortage of resources and their unbalanced supply may be some of the basic ingredients that favor nonlocal strategies.

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Luca Lussardi - 9th March 2016-

"A partial Gamma-convergence result for a family of functionals depending on curvatures"

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Biomembranes are remarkable structures with both fluid-like and solid-like properties: the main constituents are amphiphilic lipids, which have a head part that attracts water and a tail part that repels it. Because of these properties, such lipids organize themselves in micelle and bilayer structures, where the head parts shield the lipid tails from the contact with water. In a recent paper by Peletier and Röger (ARMA, 2009) a mesoscale model was introduced in the form of an energy for idealized and rescaled head and tails densities: the energy has two contributions, one penalizes the proximity of tail to polar (head or water) particles and the second implements the head-tail connection as an energetic penalization. The tickness of the structure is very small, and a full Gamma-convergence result has been proved in the same paper in the two-dimensional case: the Gamma-limit turns out to be the Euler elasitca functional for curves in the plane. The three-dimensional case is much harder and we have only partial results. In this seminar I will present the mesoscopic model proposed by Peletier and Röger, I will briefly explain how the deduction of the 2D-macroscopic model by Gamma-convergence works and then I will give some details on the 3D-case: the analysis of such a case requires deep tools from geometric measure theory, like currents and varifolds, in order to have weak notions of surfaces good for Calculus of Variations and for which a suitable notion of curvatures exists. The research project is in collaboration with Mark Peletier and Matthias Röger.

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Giovanni Eugenio Comi - 2nd March 2016 -

"Divergence-measure fields: generalizations of Gauss-Green formula with applications"

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The topic of my Master’s thesis, “Divergence-measure fields: generalizations of Gauss-Green formula with applications“, which I wrote under the supervision of Prof. K. R. Payne, concerns the study of Lp-summable vector fields whose divergence is a Radon measure, in order to achieve a generalization of the classical divergence theorem in the context of sets of finite perimeter. These fields were introduced some years ago in order to study nonlinear hyperbolic systems of conservation laws by Chen and Frid, and also the fundaments of continuum mechanics by Degiovanni, Marzocchi, Musesti and Šilhavý. We explored especially the case p=∞. The method of the proof of the Gauss-Green formula for essentially bounded divergence-measure fields is however different from the previous ones, since we adapted the techniques already developed for BV functions in Vol’pert’s work.

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Ilaria Lucardesi - 3rd December 2015-

"The wave equation on domains with cracks growing on a prescribed path"

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In this talk I analyze a scalar wave equation in a time varying domain on the form "set minus a growing crack", when the crack develops along a given path. Under suitable regularity assumptions,

I show existence, uniqueness and continuous dependence on the cracks of the weak solution of the wave equation under study. This is a joint work with Gianni Dal Maso.

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Simone Di Marino - 25th November 2015-

"BV functions and finite perimeter sets in metric measure spaces"

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We will review the BV theory in the Euclidean space, recalling briefly its relation with the geometric definition of perimeter of a set. Then we will present the recent advances in BV theory in general metric measure spaces since the general definition of Miranda, showing that it is related to the "geometric notion" of perimeter

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Federico Stra - 18th November 2015 -

"The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems"

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In this seminar I present an article by Benci and Cerami (1991) in which they show that the Dirichlet problem of the Laplacian with a power source term admits at least as many positive solutions as the Lusternik-Schnirelman category of the domain.

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Laura Cremaschi - 11th November 2015 -

"Un'introduzione al flusso di Ricci"

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Vorrei parlare in maniera piuttosto informale del flusso di Ricci e del primo lavoro di Hamilton sulle 3-varietà con curvatura di Ricci positiva. In base al tempo e al pubblico, potrei accennare al piano di Hamilton per la risoluzione della congettura di Poincaré e al lavoro di Perelman in questa direzione.

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Francois Dayrens - 28th October 2015 - 

"Reconstruction of domains from slices using phase-field approximation"

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How can we reconstruct a domain when we know it only on a finite number of slices? In the plane, given a family of segments, we search the "best" set inclosing these segments. In the usual space, the question is the same with a family of planar sections. First, by "the best set" we mean one which minimizes one of the following geometrical energies: its perimeter or the Willmore energy of its boundary. We will see that the theorical problem is not well posed but a numerical phase-field approximation seems to give an answer... an unexpected solution. For the perimeter, this solution does not "really" fit the slices. I will explain why we have this solution and how we can slightly modify the phase field scheme to have what we expect. I will give all (intuitive) definitions of concepts I use in this talk, especially the phase-field model, and we will apply this reconstruction to movie interpolation...

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Tim Laux -  21st October 2015 -

"Convergence of the thresholding schemes for geometric flows"

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The thresholding scheme, a time discretization for mean-curvature flow, was introduced by Meriman, Bence and Osher in 1992. In the talk we present new convergence results for modern variants of this scheme, in particular in the multi-phase case with arbitrary surface tensions. The first result establishes convergence towards a weak formulation of mean-curvature flow in the BV-framework of sets of finite perimeter. The proof is based on the interpretation of the thresholding scheme as a minimizing movement scheme by Esedoglu and Otto in 2014. This interpretation means that the thresholding scheme preserves the structure of (multi-phase) mean-curvature flow as a gradient flow w.r.t. the total interfacial energy. More precisely, the thresholding scheme is a minimizing movement scheme for an energy that Γ-converges to the total interfacial energy. In this sense, our proof is similar to the convergence results of Almgren, Taylor and Wang in 1993 and Luckhaus and Sturzenhecker in 1995, which establish convergence of a more academic minimizing movement scheme. Like the one of Luckhaus and Sturzenhecker, ours is a conditional convergence result, which means that we have to assume that the time-integrated energy of the approximation converges to the time-integrated energy of the limit. This is a natural assumption, which is however not ensured by the compactness coming from the basic estimates. We will also discuss new convergence results for volume-preserving mean-curvature flow and forced mean-curvature flow. --- Based on joint works with Felix Otto (MPI MIS Leipzig) and Drew Swartz (Purdue University).

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Serena Guarino Lo Bianco - 14th October 2015 - 

"The optimal reinforcement for a membrane and low congested regions"

 

We study how to rigidify an elastic membrane under the action of an exterior load f and fixed at its boundary by adding a one-dimensional reinforcement in the most efficient way; the reinforcement is described by a one-dimensional set S ⊂ Ω

which varies in a suitable class of admissible choices. We consider also the dual problem of a given region Ω where the traffic flows according to two regimes: in a region C we have a low congestion, where in the remaining part Ω∖C  the congestion is higher. The two congestion functions H1 and H2 are given, but the region C has to be determined in an optimal way in order to minimize the total transportation cost.