Seminario di Calcolo delle Variazioni & Equazioni alle Derivate Parziali

I seminari si tengono di norma di venerdì alle ore 14:30 nella Sala Conferenze "Franco Tricerri" del Dipartimento di Matematica e Informatica "Ulisse Dini" (Viale Morgagni 67/A).
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A.A. 2014 / 2015

10 luglio

ore 14:30

Kazuhiro Ishige (Tohoku University)


gioved&igrave 25 giugno

ore 14:30

Dominic Breit (Heriot Watt University, Edinburgh)
Pointwise gradient estimates for the p-Laplacian

A pointwise estimate for the gradient of solutions to the p-Laplace system with right-hand side in divergence form is established. Our contribution provides a nonlinear analogue of a classical representation formula for the Laplace system. As a consequence, a flexible, unified approach to gradient bounds for a wide class of norms is derived. In particular, well-known estimates in customary function spaces are easily recovered, and new estimates are exhibited.

12 giugno

ore 14:30

Richard James (University of Minnesota)
Materials from Mathematics

We present some recent examples of new materials whose synthesis was guided by some essentially mathematical ideas. They are materials that undergo phase transformations from one crystal structure to another, with a change of shape but without diffusion. They are hard materials, but nevertheless show liquid-like changes of microstructure under a fraction of a degree change of temperature. The underlying mathematical theory was designed to identify alloys that show low hysteresis and exceptional reversibility. The new alloys, of which Zn45Au30Cu25 is the best example so far, do show unprecedented levels of these properties, but also raise fundamental questions for theory. Some of these alloys have a strongly magnetic phase and a nonmagnetic phase. These can be used to convert heat to electricity (without the need of a separate electrical generator), and provide interesting possible ways to recover the vast amounts of energy stored on earth at small temperature difference. ( )
22 maggio

ore 14:30

Judit Abardia (Goethe Universitat, Frankfurt)
Geometric inequalities in hermitian vector spaces

In this talk we will present isoperimetric inequalities for quermassintegrals in hermitian vector spaces. Those are valuations, invariant under the unitary group and the translations of the space, and can be described as the averaging of the volume of projections over the 2 or 3-dimensional subspaces with fixed Kähler angle. The Kähler angle can be described as follows: consider the Grassmanian of real 2- or 3-planes in the hermitian space of complex dimension n, and the action of the unitary group over the Grassmanian. This action decomposes the Grassmanian into infinitely many orbits parametrized by a single real parameter, known as the Kähler angle. The isoperimetric inequalities we obtain are a consequence of a new Aleksandrov-Fenchel type inequality. The main step of the proof of the Aleksandrov-Fenchel inequality consists in associating a second order differential operator to each unitary-invariant valuation. This is a joint work with T. Wannerer.
8 maggio

ore 14:30

Debdip Ganguly (Politecnico di Torino)
Semilinear PDEs on Hyperbolic space and related problems

8 maggio

ore 15:30

Keith Miller (UC Berkeley)
Introduction to Gradient-weighted Moving Finite Elements

GWMFE is especially suited to PDE problems with sharp moving fronts. The moving nodes tend to concentrate and move with the fronts, allowing far fewer nodes and much larger timesteps. It does this by treating the solution as an evolving manifold and discretizes with an evolving piecewise linear manifold. I will explain the variational and mechanical interpretations of GWMFE, our BDF2 stiff ODE solver,and our nonlinear Krylov solver for the implicit equations. I will show 2D graphics for the Shallow Water Equations, for Normal and Vertical Motion by Mean Curvature, for the Stefan Problem for melting ice, and also an example from Neil Carlson's 3D GWMFE code. I will discuss the necessity of adding global adaptivity to our codes (insertion and deletion of nodes, flipping edges) and my largely unsuccessful search for "stabilized" versions of MFE which prevent nodes from drifting with the flow in transient advection problems.
24 aprile

ore 14:30

Gisella Croce (Université du Havre)
Su un problema ellittico a coercitivit&agrave degenere

Le equazioni ellittiche a coercitivit&agrave degenere sono state molto studiate in letteratura. In particolare si &egrave visto come la coercitivit&agrave degenere, in generale, non assicuri l'esistenza e la regolarit&agrave di soluzioni di problemi di Dirichlet. In questo seminario illustrer&ograve alcuni risultati ottenuti in collaborazione con L. Boccardo e L. Orsina sugli effetti regolarizzanti di alcuni temini di ordine inferiore.
luned&igrave 13 aprile

ore 15:00

Aljosa Volcic (Università degli Studi della Calabria)
Iterations of Steiner symmetrizations

This talk is mainly devoted to an improvement of a result due to Bianchi, Klain, Lutwak, Yang and Zhang, which says that if U is a countable and dense set of directions and if K is a convex body, then U can be ordered in a sequence so that the iteration of the Steiner symmetrizations in that order converges in the Hausdorff distance to the ball K* centered in the origin and having the same volume as K. The improvement goes in two directions. On one hand, K is allowed to be just compact (no convexity is needed), on the other hand it is shown that there exists a universal ordering of U which assures the convergence for any compact seed. At the end we will also address a result of Klain who considered iterations of Steiner symmetrizations of a convex body in a finite number of directions. This result has been extended to compact seeds in a paper by Bianchi, Burchard, Gronchi and myself. We show that an analogous result holds for measurable sets with the convergence in the L_1 distance.
10 aprile

ore 14:30

Guido De Philippis (ENS Lyon)
A direct approach to the Plateau problem

Firstly I will recall some of the attempts done in order to solve the Plateau problem and finally I will show a possible direct approach to its solution based on Preiss Theorem.
27 marzo

ore 14:30

Rolando Magnanini (Università degli Studi di Firenze)
Serrin's over-determined problem: (optimal) stability and a new proof of symmetry

The well-known Serrin's symmetry result states that, if a positive solution of a semi-linear equation in a domain D, that vanishes on its boundary, has constant gradient on the boundary, the it must be radially symmetric and D must be a ball. At the same conclusion one arrives if the gradient condition is replaced by one that requires the solution to be constant on a surface parallel (and sufficiently close) to the boundary — the parallel surface problem. I will pinpoint differences and analogies between these two problems and show that symmetry and stability of the symmetric configuration for Serrin's problem can be obtained by proving a sharp stability estimate for the parallel surface problem. The stability thus obtained for Serrin's problem is optimal (Lipschitz) and hence improves the logarithmic estimate obtained by Aftalion, Busca and Reichel and the Hœlder estimate obtained by Brandolini, Nitsch, Salani and Trombetti (for the torsional rigidity problem).
13 marzo

ore 14:30

Enrico Valdinoci (Weierstrass Institute, Berlino)
Nonlocal minimal surfaces

We present some results about the asymptotics and the regularity of the minimizers of a nonlocal perimeter functional of fractional type. In the limit, the nonlocal perimeter reduces to the classical one, but for the fractional range of the parameter some long-range interactions arise. The fractional perimeter is also related to a nonlocal partial differential equation which can be seen as the fractional analogue of the Allen-Cahn phase transition eqution, therefore a good understanding of the nonlocal minimal surfaces may provide interesting information about the interfaces of nonlocal phase transitions at a large scale.
6 marzo

ore 14:30

Luca Lussardi (Università Cattolica del Sacro Cuore, Brescia)
Su un modello variazionale per l'elasticità delle membrane cellulari

Le membrane cellulari sono costituite da un doppio strato di lipidi, a loro volta costituiti da una testa idrofila e da una coda idrofoba: questo diverso comportamento nei confronti dell'acqua circostante la membrana determina le configurazioni di equilibrio delle membrane bilipidiche. Nel 2009 Peletier e Roeger hanno proposto un modello variazionale a tre scale per descrivere il comportamento elastico delle membrane cellulari analizzando, nel caso 2D, il problema del passaggio al limite, per Gamma-convergenza, dal modello mesoscopico ad un modello macroscopico, ottenendo il funzionale elastico di Eulero. Lo scopo del seminario è quello di presentare i risultati ottenuti in collaborazione con Peletier e Roeger riguardo alla stessa problematica in 3D, la quale presenta notevoli difficolt&agrave rispetto al caso 2D, e che per essere risolta richiede l'uso di delicati strumenti di teoria geometrica delle misura, come la teoria delle correnti, dei varifolds e dei grafici generalizzati di Gauss, nozioni che consentono di dare un significato alla curvatura di una superficie anche in contesti poco regolari.
27 febbraio

ore 14:30

Luigi Ambrosio (Scuola Normale Superiore, Pisa)
BMO-like seminorms and sets of finite perimeter

In my lecture I will illustrate a recent work (in collaboration with J.Bourgain, H.Brezis, A.Figalli) on the characterization of the perimeter and of sets of finite perimeter in terms of a BMO-like seminorm, solving positively a question raised in a earlier and recent work of H. Brezis, J. Bourgain and P. Mironescu. I will more generally compare this with other distributional and non-distributional criteria for the finiteness of perimeter.
20 febbraio

ore 14:30

Rinaldo Colombo (Università degli Studi di Brescia)
Leggi di Conservazione: dalle Applicazioni alla Teoria

Negli ultimi anni, i risultati ottenuti dalla teoria delle leggi di conservazione hanno permesso di considerare nuove applicazioni di queste equazioni. Pi&ugrave recentemente, queste applicazioni hanno posto nuove domande a cui la matematica, a vari livelli, deve cercare di rispondere. In questo seminario verranno presentati alcuni di questi problemi, suggeriti ad esempio dal traffico veicolare e da un'applicazione di tipo industriale.
13 febbraio

ore 14:30

Filomena Pacella (Università di Roma "La Sapienza")
Blow up in finite time for some semilinear heat equations

We consider semilinear heat equations with power nonlinearities in bounded domains with Dirichlet boundary conditions and show some properties of the set of initial data for which the corresponding solutions blow up in finite time. A particular phenomenun will be pointed out, taking initial data close to some sign changing stationary solutions of the associated semilinear elliptic problem. The cases considered concern power nonlinearities with exponents close to some limit value. Some of the results presented have been obtained in collaboration with F. Dickstein, V. Marino and B. Sciunzi.
30 gennaio

ore 14:30

Flaviana Iurlano (HCM Bonn)
Approssimazioni phase field di modelli di frattura

Nella letteratura degli ultimi anni ampio spazio &egrave stato dedicato alla derivazione di modelli di frattura fragile a partire da modelli pi&ugrave regolari, come modelli di danneggiamento e phase field, principalmente in termini di Gamma-convergenza. Il primo risultato in questo senso &egrave stato ottenuto da Ambrosio e Tortorelli per l'energia di Griffith, successivamente generalizzato in diverse direzioni con scopi differenti. Ci soffermeremo su un recente risultato per le energie di frattura coesiva alla Barenblatt nel caso di taglio antipiano. L'estensione al caso generale dell'elasticit&agrave linearizzata in dimensione n resta tuttora un problema aperto. Discuteremo di alcune questioni utili al fine di perseguire tale scopo.
16 gennaio

ore 14:30

Maria Colombo (Scuola Normale Superiore, Pisa)
Regolarit&agrave di problemi variazionali a due fasi

Negli anni '80, Zhikov ha introdotto alcuni funzionali integrali che modellizzano materiali compositi non omogenei, ottenuti mescolando due differenti materiali semplici. Un funzionale modello &egrave un integrale variazionale, il cui integrando varia tra due diversi tipi di fasi ellittiche, in base ad un coefficiente continuo che dipende dalla posizione. Se le fasi ellittiche sono distanti in modo quantificato, i minimi possono essere discontinui. Altrimenti, mostriamo la regolarit&agrave ottimale per i minimi, ovvero la continuit&agrave holderiana del gradiente (lavoro in collaborazione con Paolo Baroni e Rosario Mingione).
19 dicembre

ore 15:00
Nicola Zamponi (Vienna University of Technology)
Analytical study of degenerate cross-diffusion population models with volume filling

We present a broad class of population models with nonlinear degenerate cross-diffusion structure and volume-filling. The existence of bounded nonnegative weak solutions to the system is shown by semi-discretizing the system in time and by adding a suitable regularizing term. A discrete entropy inequality allows to obtain gradient estimates which are uniform with respect to the time step and the regularization parameter. In order to overcome the difficulties arising from the degeneracy and the nonlinearity, a known compactness result [5, Lemma 13] is generalized, allowing to prove strong convergence results for the variables of interest. The positivity of the solvent density is obtained under additional assumptions. The convergence of the solution to the steady state as t ? ? is proved by means of suitable convex Sobolev inequalities and an ad-hoc application of Egorov?s theorem for measurable functions on finite-measure sets. The uniqueness of bounded weak solutions for the system in a particular case is shown by applying the H^{-1} method and by proving a suitable inequality for the relative entropy between two solutions, which is a consequence of the convexity property of the Fisher information.
12 dicembre

ore 14:30

Ettore Minguzzi (Università degli Studi di Firenze)
The area theorem and its applications

A celebrated result of mathematical relativity states that Black Holes have a non-decreasing area. This result was proved by Hawking under tacit differentiability assumptions and suggests to identify the area with the entropy of these astrophysical objects. By using tools from geometric measure theory I show how to prove a strong version of Hawking's area theorem. As a corollary to this stronger version, I prove that compact Cauchy horizons are smooth and cannot form under generic conditions (a case of cosmic censorship) and that topology change or time machines cannot be realized in general relativity.
5 dicembre

ore 14:30

Federica Dragoni (Cardiff University)
Overview on convex functions in sub-Riemannian geometries.

28 novembre

ore 14:30

Paolo Maria Mariano (Università degli Studi di Firenze)
Variational description of stress constraints in simple bodies undergoing large strains

21 novembre

ore 14:30

Nadia Clavero (Universitat de Barcelona, Spagna)
Optimal Gagliardo-Nirenberg type estimates

marted&igrave 11 novembre

ore 14:30

Teemu Lukkari (University of Jyvaskyla, Finland)
Perron's method for the porous medium equation

7 novembre

ore 14:30

Manuel del Pino (Universidad de Chile)
Green's function and infinite time bubbling in the semilinear heat equation at the critical Sobolev exponent

We discuss some new results on globally defined in time positive solutions of the semilinear heat equation with critical power nonlinearity and Dirichlet boundary conditions in a bounded domain. For any given number k we can find a solution that, as time grows, blows up exactly at k points of the domain with a bubbling profile that can be precisely computed. This is joint work with Monica Musso.
24 ottobre

ore 14:30

Berardo Ruffini (Institut Fourier, Grenoble)
Non-local isoperimetric problems.

After recalling some preliminary results and definitions needed for a clear comprehension of the argument, we show a quantitative-type isoperimetric inequality for fractional perimeters where the deficit of the t-perimeter controls from above that of the s-perimeter, with s smaller than t. To do this we consider a problem of independent interest: we charachterize the volume constrained minimizers of a free energy problem given by the difference of the t-perimeter and the s-perimeter. Finally we link this problem with a mixed non-local isoperimetric problem which generalizes the classical isoperimetric inequality.
10 ottobre

ore 14:30

Eugene Stepanov (St. Petersburg State University)
Measurable vector fields, rectifiable curves and flows of measures

A smooth vector field (say, over a mainfold) may be defined either as a linear operator on the algebra of smooth functions satisfying Leibniz rule, or, equivalently, as a smooth field of directions of curves (i.e. "vectors") at every point. The first notion easily generalizes to what is known as measurable vector fields introduced by N. Weaver. These vector fileds can in fact be identified with one-dimensional metric currents of Ambrosio and Kirchheim. We show that an identification similar to the smooth case is valid for a large class of measurable vector fileds (but not all of them) and study the analogues of integral curves and ODE's produced by such vector fields as well as the flows of measures generated by them.
3 ottobre

ore 14:30

Maarten V. De Hoop (Purdue University)
A Geometric Inverse Problem connected to Microseismicity

26 settembre

ore 14:30

Lenka Slavikova (Charles University, Prague)
Higher order Sobolev embeddings and isoperimetric inequalities

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