Seminario di Analisi a.a. 2010/11

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Abstract: in questo seminario si ricorderanno alcune ben note disuguaglianze isoperimetriche relative agli autovalori dell'operatore di Laplace, sia nel caso di dato al bordo di tipo Dirichlet che di tipo Neumann. Si mostrera' come esse possano essere migliorate, rendendole quantitative: in particolare, si mostrera' la possibilita' di stimare la vicinanza (in un qualche senso) di un oggetto all'insieme isoperimetrico, in termini della vicinanza del relativo autovalore. Alcuni dei risultati quivi presentati fanno parte di un lavoro in collaborazione con Aldo Pratelli (Pavia).

Abstract: Nel 1978, De Giorgi chiese se le soluzioni limitate e monotone dell'equazione di Allen-Cahn dipendessero da una sola variabile euclidea - almeno in dimensione minore o uguale a otto. Discuteremo dello stato dell'arte di questo problema e di recenti sviluppi in problemi collegati.

Recentemente nello studio di fenomeni legati alle transizioni di fase è emerso l'interesse per la definizione di funzionali di tipo perimetro non locali. Lo scopo del seminario è presentare alcuni recenti risultati ed in particolare mostrare un risultato di Gamma convergenza al perimetro "classico".

Abstract: For second order semilinear equations of mixed elliptic-hyperbolic type on bounded domains in the plane, we will present some recent progress on the question of existence/nonexistence of non trivial weak solutions that satisfy a homogeneous boundary condition of Dirichlet type on all or part of the boundary. In particular, a better undrstanding of what consitutes critical growth for such problems will be discussed. The results to be presented are part of an ongoing collaboration with Daniela Lupo (Politecnico di Milano), Dario Monticelli (universita' di Milano) and Nedyu Popivanov (University of Sofia).

We derive the sharp constants for the inequalities on the Heisenberg group H^n whose analogues on Euclidean space R^n are the well known Hardy-Littlewood-Sobolev inequalities. Only one special case had been known previously, due to Jerison-Lee more than twenty years ago. From these inequalities we obtain the sharp constants for their duals, which are the Sobolev inequalities for the Laplacian and conformally invariant fractional Laplacians. By considering limiting cases of these inequalities sharp constants for the analogues of the Onofri and log-Sobolev inequalities on H^n are obtained. The methodology is completely different from that used to obtain the R^n inequalities and can be used to give a new, rearrangement free, proof of the HLS inequalities.

Abstract: We propose a variational approach to quantitative isoperimetry. Our method is based on a penalization argument and on the regularity theory for quasiminimizers of the perimeter. Two remarkable applications of the method will be presented: first, a new proof of the quantitative isoperimetric inequality in R^n; second, the proof of a conjecture posed by Hall in 1992.

Abstract: Proporremo una versione "quasi uniforme" del principio di antimassimo, ottenuta recentemente con Genni Fragnelli, mostrandone alcune applicazioni.

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.

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Abstract: In questo seminario mi propongo di presentare lo sviluppo e i progressi recenti della teoria del controllo ottimale con funzionali quadratici per sistemi che descrivono problemi al contorno e ai valori iniziali per Equazioni a Derivate Parziali (EDP) con controllo sulla frontiera o puntuale. Di specifico interesse sono sistemi di EDP accoppiate di tipo diverso quali sistemi termoelastici, sistemi che descrivono interazioni tra onde acustiche e vibrazioni elastiche, modelli per le interazioni fluido-solido. Un avanzamento importante nello studio di tali problemi si e' avuto di recente con l'introduzione di una nuova formulazione astratta, capace di individuare proprieta' di regolarita' distintive della dinamica `ibrida' sufficienti a garantire la risolubilita' dei problemi di controllo ottimale associati, assieme alla buona positura delle equazioni di Riccati corrispondenti, ove invece le teorie classiche falliscono. Il ruolo speculare ma centrale della regolarita' delle tracce delle soluzioni del sistema di EDP "libero" (cioe' in assenza di controllo) verra' discusso ed illustrato.

Abstract: In this talk we address some geometric optimization problems with convexity constraints. We are specially interested in problems whose solutions are singular, i.e. polygons. We will give an approach to study such questions, which is general for a wide class of problems, at least in two dimensions. We will illustrate this approach on several examples, including a geometric problem related to the reverse isoperimetric inequality and also the Mahler conjecture.

Abstract: The construction of the center manifold is probably the most technical and difficult part of Almgren's celebrated work on the regularity of area-minimizing currents with codimension larger than $1$. Roughly speaking, in the neighborhood of a point of branching, the center manifold is close to the average of the sheets. As pointed out by Almgren in the introduction to his work, a remarkable corollary of his construction is a direct proof (i.e. without using the nonparametric theory of minimal surfaces) of $C^{3,\alpha}$ regularity in neighborhoods of points where no branching occurs. In this talk we show how to prove this corollary in a rather straightforward way, hoping that this will be the starting step of a much cleaner proof of Almgren's full result.

Abstract: In this talk we address some features of variational integrals of linear growth. A model case is given by the minimization problem for the integral $$\int \sqrt{1 + |\nabla w|^2} dx$$ in Dirichlet classes of vector-valued functions w. After introducing the concept of generalized minimizers, we briefly comment on their existence and then focus on a recent result, namely their uniqueness (up to additive constants). Providing several examples we finally discuss in more detail the phenomenon of non-uniqueness, which is closely related to the possible non-attainment of the boundary values. All results were obtained in collaboration with T. Schmidt (Erlangen).

Abstract: In this seminar we will discuss the inverse problem of detecting a single body immersed in a bounded domain, filled with a fluid obeying the Stokes equations, from a single measurement of forces and velocity on a portion of the boundary of the domain. We will first state the problem and discuss its ill-posedness. We will then state the stability result and sketch the steps taken in the proof. We will also discuss the adaptations needed in order to generalize the result to the case where the fluid obeys the nonlinear stationary version of Navier-Stokes equations.

Sunto: Il problema di Minkowski classico richiede di trovare un corpo convesso, cioe' un insieme convesso e compatto, con frontiera regolare, assegnata la curvatura di Gauss in funzione della normale esterna al bordo. Questo problema ammette una formulazione debole nella quale il dato e' la cosiddetta misura d'area di un corpo convesso. Questa misura puo' essere interpretata come la variazione prima del volume rispetto a certe perturbazioni del corpo. Lo scopo di questo seminario e' quello di sviluppare questo punto di vista e di descrivere problemi analoghi in cui il volume viene sostituito da funzionali classici del calcolo delle variazioni, come la capacita', il primo autovalore dell'operatore di Laplace con dato di Dirichlet e la rigidita' torsionale.

Sunto: Considerano i minimi per il funzionale geometrico $\lamba$ Area - Perimetro, nella classe di insiemi convessi contenuti in un anello fissato, si dimostrano alcune disuguaglianze riguardanti l'area, il perimetro e l'inradius o il circumradius di insiemi convessi.

Abstract: In the study of a priori bounds for systems of stationary Schrödinger systems - as arising e.g. in the Hartree-Fock theory of a mixture of Bose-Einstein condensates - one is led, via a rescaling procedure, to analyze the solution set of cubic elliptic systems either in the entire space or in the half space. We will focus on existence and nonexistence results in the case where the system is not cooperative. Here the structure of the solution set depends in a subtle way on parameters, and standard methods like the moving plane method cannot be used.

Abstract: In the simplest form, our result characterizes the bounded, divergence-free vector fields on the plane such that the Cauchy problem for the associated continuity (or transport) equation has a unique bounded solution in the sense of distribution. Unlike previous results in this directions (cf. Di Perna-Lions, Ambrosio, etc.), the proof does not rely on regularization, but rather on a dimension-reduction argument. Note that the "good" vectorfields are not characterized in terms of function spaces (i.e., by inequalities), but by a qualitative property which is completely non-linear in character. These results have been obtained in collaboration with Stefano Bianchini (SISSA, Trieste) and Gianluca Crippa (University of Parma).

Abstract: Presenteremo alcuni nuovi risultati di locale limitatezza delle soluzioni di sistemi ellittici quasilineari sotto ipotesi di crescita anisotropa e p-q. Daremo anche una panoramica dello stato dell'arte della regolarit&agrave sotto ipotesi di crescita non-standard.

Abstract: The Brascamp--Lieb Poincar&eacute type inequality provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the L^2 norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto have recently proved a variant of this in one dimension with the two L^2 norms replaced by L^1 and L^\infty norms, but only for R. We prove a generalization of both by extending these inequalities to L^p and L^q norms and on R^n, for any n\geq 1. We also prove an inequality for integrals of divided differences of functions in terms of integrals of their gradients, with an interesting relation to both the Brascamp-Lieb inequality and the KLS conjecture. This is joint work with Dario Cordero-Erausquion and Elliott Lieb.

Abstract: In this talk I will present some recent results on higher integrability, "nondoubling" Muckenhoupt weights and regularity of minimal currents. In particular, I will show an equivalent condition to the higher integrability with increasing supports which is a generalization of the Muckenhoupt weights holding true also for nondoubling measures. This result is in turn used to deduce a new a priori estimate for minimal currents in any codimension. Time permitting, some connection with the problem of approximation of minimal currents will be also highlighed.

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Verranno discussi alcuni recenti risultati riguardanti una congettura di De Giorgi, che permette di ottenere problemi di Cauchy per equazioni delle onde non lineari (defocusing, con non linearità arbitraria di tipo potenza) come limite di problemi di minimizzazione per funzionali convessi del Calcolo delle Variazioni, dipendenti da un parametro. Saranno anche discussi alcuni problemi aperti e diverse possibili estensioni.

L'oggetto del seminario &egrave lo studio della relazione tra k-quasiconvessit&agrave e quasiconvessit&agrave, per ogni k >= 2. Mostreremo che sotto opportune ipotesi una funzione strettamente k-quasiconvessa e con p crescita (p>1) &egrave la restrizione all'insieme dei k-tensori simmetrici di una funzione quasiconvessa con la stessa crescita. Questo permette di dedurre risultati di semicontinuit&agrave per problemi variazionali di ordine k utilizzando teoremi gi&agrave noti per problemi di ordine 1. I risultati esposti generalizzano un lavoro di Dal Maso, Fonseca, Leoni e Morini riguardante il caso k=2.

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.

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It often useful to approximate the symmetric decreasing rearrangement by a sequence of simpler rearrangements, such as polarization (two-point rearrangement), Steiner symmetrization, or the Schwarz rounding process. In this talk, I will discuss recent results with Marc Fortier on the convergence of random sequences of rearrangements. We derive conditions under which random sequences of polarizations converge almost surely to the symmetric decreasing rearrangement. The parameters for the polarizations are independent random variables whose distributions may be far from uniform. In the special case of i.i.d. sequences, there is almost sure convergence even for polarizations chosen at random from small sets. These statements about polarization allow us to improve the existing convergence results for Steiner symmetrization. In particular, we show that full rotational symmetry i can be achieved by Steiner symmetrization along a random sequence chosen from finite sets of directions that satisfy an explicit non-degeneracy condition. Finally, we construct examples for dense sequences of directions such that the corresponding Steiner symmetrizations do not converge.