Seminario di Analisi Matematica

• giovedì 7 luglio 2022, 14h30. Aula Tricerri e streaming. Francesca Bucci, Università di Firenze, Riccati theory in the realm of PDEs: an overview and recent advances in the optimal control of evolution equations with memory
  Abstract: The well-posedness of Riccati equations plays a central role in the study of the optimal control problem with quadratic functionals for linear partial differential equations (PDEs). Indeed, it allows the synthesis of the optimal control by solving the Riccati equation corresponding to the minimization problem, and then the closed-loop equation. In this lecture I will recall the principal steps of an established functional-analytic approach to the linear-quadratic problem for infinite-dimensional systems representing PDEs with distributed or boundary control. The challenges (and possible failures) stemming from the presence of unbounded control operators, combined with a hyperbolic or composite dynamics, will be highlighted, along with the PDE interpretation of certain control-theoretic conditions. I will finally outline how the said approach proves successful even in the case of certain evolution equations with finite memory, thereby providing a first extension (to the realm of PDEs) of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context.
  (Talk based on ongoing joint work with P. Acquistapace (Pisa), besides past work with Acquistapace and I. Lasiecka (Memphis).)

• 12 novembre 2021, 14h30. Aula 102 DiMaI e streaming. Andrea Colesanti, Università di Firenze, Funzionali additivi su insiemi convessi e funzioni convesse
  Sunto: Nella prima parte del seminario verrà data un'idea della teoria dei funzionali finitamente additivi (chiamati comunemente valutazioni), sugli insiemi convessi. La seconda parte verrà dedicata ad alcuni recenti sviluppi della teoria delle valutazioni, che hanno portato ad una sua parziale estensione a spazi funzionali, e in particolare a spazi di funzioni convesse.


• 26 novembre 2021, 14h30. Aula 102 DiMaI e streaming. Emanuele Spadaro, Sapienza Università di Roma, Sul comportamento asintotico del flusso per curvatura media a volume costante
  Abstract: In questo seminario discuterò alcuni risultati sul comportamento asintotico delle soluzioni approssimate del flusso per curvatura a volume costante. Questa è una variante del più famoso flusso geometrico per curvatura media, che descrive l'evoluzione di sistemi a due fasi governate dalla tensione superficiale per le quali si ha conservazione della massa. Come conseguenza di una versione quantitativa del teorema di Alexandrov sulla superfici a curvatura media costante, mostrerò la convergenza di un flusso approssimato alle configurazioni stazionarie fatte da componenti sferiche della stessa dimensione.


• 03 dicembre 2021, 14h30. Solo in streaming. Raffaele Grande, Cardiff University, Stochastic control problems and evolution by horizontal mean curvature flow
  Abstract: The horizontal mean curvature flow (HMCF) is widely used in neurogeometry and in image processing (e.g. Citti Sarti model). It represents, informally, the contracting evolution of a hypersurface embedded in a particular geometrical setting, called sub-Riemannian geometry, in which only some curves (called horizontal curves) are admissible by definition. This may lead the existence of some points of the hypersurface called characteristic points in which is not possible to define the horizontal normal. In order to avoid this problem, it is possible to use the notion of Riemannian approximation of a sub-Riemannian geometry applied to the horizontal mean curvature flow. I will show the connection between the evolution of a generic hypersurface in this setting and the associated stochastic optimal control problem (as stated e.g. by Cardaliaguet, Buckdahn and Quincampoix and Soner and Touzi in the Euclidean and Riemannian case and Dirr, Dragoni and von Renesse in the sub-Riemannian setting). To conclude, I will show some results which I have found in collaboration of N. Dirr and F. Dragoni about asymptotic optimal controls in the Heisenberg group (in the horizontal and the approximated case), about the convergence of approximated solutions to the horizontal solutions.


• 17 dicembre 2021, 14h30. Aula 202 al II piano DiMaI e streaming. Nella Rotundo, Università di Firenze, On the existence results of solutions of a drift diffusion system modeling the lateral photovoltage scanning method
  Abstract: The lateral photovoltage scanning method (LPS) detects doping inhomogeneities in semiconductors such as silicon and germanium in a cheap, fast, and nondestructive manner. LPS relies on bulk photovoltaic effects and thus can detect any physical quantity affecting band profiles of the sample. We model this technique by coupling a drift-diffusion model enriched by an additional sourcing term in the continuity equations, combined with an implicit boundary condition. The model can be seen as a special case of coupled models for circuits and semiconductors. In this talk, I will first describe classical analytical techniques used in this context and then I will show which kind of difficulties arise in proving similar results in the LPS case. I will also discuss possible strategies of reduction of the LPS model in order to get existence results in the LPS case.


• 21 gennaio 2022, 14h30. Aula 102 DiMaI e streaming. Anna Kausamo, Università di Firenze, The Monge problem in optimal mass transportation: from two to many marginals
  Abstract: In this talk I will introduce the topic of optimal mass transportation (OT) concentrating on the problem of the existence of a deterministic solution (The Monge Problem). I will start from the standard two-marginals framework and then move on to the multi-marginal case where the nature of the Monge problem - and its level of difficulty - changes completely. The results presented are joint work with Tapio Rajala and Augusto Gerolin.


• 4 febbraio 2022, 14h30. Aula 102 DiMaI e streaming. Riccardo Durastanti, Università di Napoli Federico II, Spreading equilibria under mildly singular potentials: pancakes versus droplets
  Abstract: We study minimizers of a functional modeling the free energy of thin liquid layers over a solid substrate under the combined effect of surface, gravitational, and intermolecular potentials. When the latter ones have a mild repulsive singularity at short ranges, minimizers are compactly supported and display a right microscopic contact angle. Depending on the form of the potential, the macroscopic shape can either be droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient. These results generalize, complete, and give mathematical rigor to de Gennes' formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed. This is a joint work with Lorenzo Giacomelli.


• 18 febbraio 2022, 14h30. Aula 102 DiMaI e streaming. Carmen Nunez, Universidad de Valladolid, Rate-induced critical transitions and saddle-node bifurcation for quadratic nonautonomous ODEs
  Abstract: Several complex systems in nature and society have been proved susceptible to abrupt, large and irreversible transitions in their behaviour, as a consequence of relatively small changes in parameters describing external conditions. These often unexpected changes are commonly referred to as tipping points (or critical transitions), and they have been reported by applied scientists in various contexts, including epileptic seizures, ecology, earthquakes, and climate. An in-depth analysis of nonautonomous bifurcations of saddle-node type for scalar nonautonomous differential equations x'=-x^2+q(t) x+p(t) where q: R---> R and p: R--> R are bounded and uniformly continuous, is fundamental to explain the absence or occurrence of rate-induced tipping points for the differential equation y' =(y-(2/\pi)\arctan(ct))^2+p(t) as the rate c varies on [0,\infty). A classical attractor-repeller pair, whose existence for c=0 is assumed, may persist for any c>0, or disappear for a certain critical rate c=c_0, giving rise to rate-induced tipping. A suitable example demonstrates that this tipping phenomenon may be reversible.


• 25 febbraio 2022, 14h30. Aula 102 DiMaI e streaming. Aldo Pratelli, Università di Pisa, Some recent results on the minimization of energies of attractive-repulsive type
  Abstract: The celebrated liquid drop model by Gamow states that the shape E of a charged particle should minimize the sum between the perimeter of E and the double integral over E\times E of 1/|x-y|. The first term has an attractive effect and the second one, which is of non-local form, a repulsive one. In the last few years a very high number of generalisations have been proposed. In particular, the attractive term has been replaced by a fractional perimeter, or by a non-local attractive term given by a positive power of |y-x|; and the repulsive term has been replaced by any meaningful negative power, or by even more general terms. Moreover, also the class in which looking for a minimizer has been extended, namely, not only sets but more in general L^1 functions, or even just measures. In this talk we will try to give a quick but complete overview of the problem and of the main results of last ten years. We will conclude by presenting some very recent recents on these questions.


• 4 marzo 2022, 14h30. Aula 102 DiMaI e streaming. Luca Bisconti, Università di Firenze, On the convergence rates for the three-dimensional filtered Boussinesq equations
  Abstract: In this talk we consider a LES-turbulence model for the approximation of large scales of the 3D Boussinesq equations. We analyze the convergence rate, in appropriate norms, of the unique regular weak solution of the regularized Boussinesq equations towards a weak solution of the original Boussinesq system. In so doing we first estimate the difference between the weak solutions of the original system and that of the regularized one, the so-called approximation error, in terms of consistency errors, then we focus and provide sharper estimates for the latter. These errors arise from the approximation scheme when inserted into the original system; in particular we give appropriate bounds on the model's consistency errors following some recent and specific results on this topic.


• 11 marzo 2022, 14h30. Aula 202 al II piano DiMaI e streaming. Francesco Pediconi, Università di Firenze, Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
  Abstract: In this talk, we study Sobolev regularity of solutions to nonlinear second order elliptic PDEs with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue scale. Our method is based on an integral refinement of the Bernstein method, and leads to "semilinear Calderón-Zygmund" type results. This is joint work with A. Goffi.


• giovedì 17 marzo 2022, 12h30. Aula 203 DiMaI e streaming. Adriana Garroni, Sapienza Università di Roma, Derivation of surface tension of grain boundaries in polycystals
  Abstract: Inspired by a recent result of Lauteri and Luckhaus, with derive, via Gamma convergence, a surface tension model for polycrystals in dimension two. The starting point is a semi-discrete model accounting for the possibility of having crystal defects. The presence of defects is modelled by incompatible strain fields with quantised curl. In the limit as the lattice spacing tends to zero we obtain an energy for grain boundaries that depends on the relative angle of the orientations of the two neighbouring grains. The energy density is defined through an asymptotic cell problem formula. By means of the bounds obtained by Lauteri and Luckhaus we also show that the energy density exhibits a logarithmic behaviour for small angle grain boundaries in agreement with the classical Shockley Read formula. The talk is based on a paper in preparation in collaboration with Emanuele Spadaro.


• 25 marzo 2022, 14h30. Aula 202 DiMaI e streaming. Ilaria Lucardesi, Università di Firenze, On the maximization of the first (non trivial) Neumann eigenvalue of the Laplacian under perimeter constraint
  Abstract: In this talk I will present some recent results obtained in collaboration with A. Henrot and A. Lemenant (both in Nancy, France), on the maximization of the first (non trivial) Neumann eigenvalue, under perimeter constraint, in dimension 2. Without any further assumption, the problem is trivial, since the supremum is $+\infty$. On the other hand, restricting to the class of convex domains, the problem becomes interesting: the maximum exists, but neither its value nor the optimal shapes are known. In 2009 R.S. Laugesen and B.A. Siudeja conjectured that the maximum among convex sets should be attained at squares and equilateral triangles. We prove that the conjecture is true for convex planar domains having two axes of symmetry.


• Segnaliamo il primo "Colloquio di Matematica del Dini" organizzato da Daniele Angella, Matteo Focardi, Eugenio Giannelli:
  giovedì 31 marzo 2022, 14h30. Aula 203 DiMaI e streaming. Alfio Quarteroni, PoliMi e EPFL, Physics-based and data-driven mathematical models for the simulation of the heart function


• 8 aprile 2022, 14h30. Aula 202 DiMaI e streaming. Francisco Marín Sola, Universitad de Murcia, On extensions of Grünbaum's inequality
  Abstract


• 29 aprile 2022, 14h30. Aula 202 DiMaI e streaming. Bozhidar Velichkov, Università di Pisa, Free boundary clusters with two phases
  Abstract: We will discuss a two-phase free boundary problem in which the two state functions are not vanishing on the free interface between the two phases. The talk is based on joint works with Serena Guarino Lo Bianco and Domenico Angelo La Manna.


• 6 maggio 2022, 14h30. Aula 202 DiMaI e streaming. Abbas Moameni, Carleton University (Ottawa), On the m-twist condition, structure and the uniqueness of optimal transport plans
  Abstract: We shall introduce the notion of m-twist condition for the optimal transportation problems. This notion together with Kantorovich duality provide an effective tool to study and describe the support of optimal plans for the mass transport problem involving general cost functions. We also establish a criterion for the uniqueness.


• 13 maggio 2022, 12h30. Aula 203 DiMaI e streaming. Richard Gardner, Western Washington University, Geometric tomography: An update on open problems, including the hyperplane and Mahler conjectures
  Abstract: Geometric tomography aims to retrieve information about a geometric object (such as a convex body, star body, finite set, etc.) from data concerning its intersections with planes or lines and/or projections (i.e., shadows) on planes or lines. The topic offers the researcher many interesting open problems, and we shall describe the current status of several of them.
  There is a significant overlap between geometric tomography and analytical convex geometry, and in particular these subjects share two notoriously difficult open problems: The hyperplane conjecture (or slicing problem) of Bourgain, and Mahler's conjecture. Both problems have generated a large literature in which connections with many other open problems from a number of different areas in mathematics have been observed, for example, analysis, probability, information theory, number theory, and computational, Finsler, integral, stochastic, and symplectic geometry. We shall attempt to provide an overview of these connections.
  The talk will be aimed at a general audience.


• giovedì 26 maggio 2022, 14h30. Aula 203 DiMaI e streaming. Jyotshana V. Prajapat, University of Mumbai, Convex sets in Heisenberg group
  Abstract: I will discuss possible definitions of convex sets in the Heisenberg group and classify that the geodetically convex subsets of the Heisenberg group are either emptyset, singleton sets, arcs of geodesics or the whole space. We give a proof extending the result of Monti-Rickly to higher dimensions.


• 27 maggio 2022, 14h30. Aula 202 DiMaI e streaming. Diego Berti, Università di Firenze, A regularity criterion for a 3D tropical climate model with damping
  Abstract: In the context of fluid mechanics, in this talk we consider the Tropical Climate Model (TCM) in three spatial dimensions. TCM consists of a system of three Navier Stokes-like equations. The unknowns here are two vector fields (standing for the barotropic mode and the first baroclinic mode of the velocity) as well as a scalar function (the temperature). In the case under examination, the equations are also equipped with damping on the barotropic mode and on the first baroclinic mode of the velocity.
  For this system, we give a regularity criterion thanks to which the local smooth solution of the associated Cauchy problem, can actually be extended globally in time. The criterion is given in terms of specific homogeneous Besov spaces.
  This talk is based on a joint work with L.Bisconti (Firenze) and D.Catania (Brescia and eCampus).


• 10 giugno 2022, 14h30. Aula 201 DiMaI e streaming. Lorenzo Baldassari, Rice University (Houston, Texas), Computing the electromagnetic field scattered by a metallic nanoparticle
  Abstract: We study the electromagnetic field scattered by a metallic nanoparticle with dispersive material parameters placed in a homogeneous medium in a low-frequency regime. We obtain a modal approximation of the scattered field in the frequency domain. The poles of the expansion correspond to the eigenvalues of a singular boundary integral operator and are shown to lie in a bounded region near the origin of the lower-half complex plane. Finally, we show that this modal representation gives a very good approximation of the low-frequency part of the field in the time domain. We present numerical simulations in two dimensions to corroborate our results. We illustrate the usefulness of our method on the super-localisation of a point-like emitter in a resonant environment.


• 10 giugno 2022, 15h30. Aula 201 DiMaI e streaming. Giorgio Poggesi, University of Western Australia, Analysis meets geometry -- on some geometric properties in PDEs
  Abstract


• 17 giugno 2022, 14h30. Aula 201 DiMaI e streaming. Dmitry Ryabogin, Kent State University (Ohio, USA), Ulam's problem 19 from the Scottish Book and related problems
  Abstract: Ulam's Problem 19 from the Scottish Book asks: is a solid of uniform density which will float in water in every position a sphere?
  Assuming that the density of water is 1, one can show that there exists a strictly convex body of revolution K\subset R3 of uniform density 1/2 which is not a Euclidean ball, yet floats in equilibrium in every orientation.
  We will discuss this and related problems suggested by Croft, Falconer and Guy.


• giovedì 23 giugno 2022, 14h30. Aula 201 DiMaI e streaming. Eduardo Muñoz Hernández, Universidad Complutense de Madrid, Nodal solutions in a general Moore-Nehari problem: a dynamic and bifurcation approach
  Abstract

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