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. 2019 / 2020

--::: PROSSIMO SEMINARIO: :::--

I seminari sono sospesi fino a quando non riprenderanno le attività del Dipartimento.

----:: Seminari passati: ::----

28 febbraio

14:30

Andreas Bernig (Goethe-Universität Frankfurt)

A Chern-Gauss-Bonnet theorem for metrics with changing signature

Abstract: The classical Chern-Gauss-Bonnet theorem gives a relation between the geometry and the topology of a Riemannian manifold. More precisely, the Euler characteristic is given as integral over some polynomial expression in the curvature tensor. We generalize this theorem to manifolds endowed with a generic symmetric $(0,2)$-tensor which may have changing signature. Examples of such manifolds are Hessian-type metrics; they also appear in physics (general relativity, cosmology, quantum gravity, optical metamaterials). Our main theorem states that for such a manifold, the Euler characteristic is obtained by integrating some natural distribution over the manifold. The proof is based on a pseudo-Riemannian version of the Weyl principle and uses Alesker's theory of valuations on manifolds. This is joint work with Dmitry Faifman, Univ. Montréal and Gil Solanes, UA Barcelona.

21 febbraio

14:30 e 15:30

Frank Duzaar (FAU Erlangen-Nürnberg)

Integral convexity and parabolic systems

Verena Bögelein (Universität Salzburg)

Hölder continuity for a homogenous doubly nonlinear equation

Abstract Duzaar

Abstract Bögelein

14 febbraio

14:30

Giulio Ciraolo (Università di Milano)

Symmetry results for critical p-Laplace equations

Abstract

24 gennaio

14:30

Alberto Roncoroni (Università di Firenze)

Serrin's type rigidity results

Abstract: The talk deals with overdetermined elliptic boundary value problems for bounded domains in the Euclidean space and, more in general, in Riemannian manifolds. Starting from the well-known Serrin's rigidity result, an analogue result for domains inside convex cones of the Euclidean space will be considered. This part is based on a joint work with G. Ciraolo and on a recent project in collaboration with G. Ciraolo, P. Sicbaldi and B. Sirakov. Moreover, rigidity results and counterexamples in space forms (the hyperbolic space and the sphere) will be shown. In the context of more general Riemannian manifolds (such as model manifolds and warped product manifolds), several rigidity results for overdetermined problems will be considered. This part is based on a joint work with A. Farina.

17 gennaio

14:30, AULA 1

Antonia Passarelli di Napoli (Università di Napoli Federico II)

Regolarità per minimi di una classe di funzionali con convessità degenere

10 gennaio 2020

14:30 e 15:30

Toru Kan (Osaka Prefecture University)

A remark on the behavior of solutions of the Allen-Cahn equation in the whole plane

Michiaki Onodera (Tokyo Institute of Technology)

Hyperbolic solutions to Bernoulli's free boundary problem

Abstract Toru Kan: We are concerned with the behavior of the interface (level curve) of solutions of the Allen-Cahn equation in the whole plane. It is known that the interface of a radially symmetric solution moves slightly slower than that of a planar traveling wave solution. More specifically, the distance of the interfaces grows logarithmically as time goes to infinity. In this talk, we show that the distance between the interface of some solution and that of a planar traveling wave solution can grow polynomially.

Abstract Michiaki Onodera: Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are ``stable'' solutions and tractable by variational methods and maximum principles, while hyperbolic solutions are ``unstable'' solutions of which the qualitative behavior is less known. I will present a joint work with Antoine Henrot (Institute Elie Cartan), in which we introduce a new implicit function theorem based on parabolic maximal regularity, applicable to problems with loss of derivatives. Clarifying the spectral structure of the corresponding linearized operator by harmonic analysis, we prove the existence of foliated hyperbolic solutions as well as elliptic solutions in the same regularity class.

13 dicembre

14:30 e 15:30

Lisa Beck (Augsburg University)

On the structure of BV minimizers of linear growth functionals

Nikos Katzourakis (University of Reading)

On the existence and uniqueness of vectorial absolute minimisers in Calculus of Variations in L-infinity

Abstract Beck

Abstract Katzourakis: Calculus of Variations in the space L-infinity has a relatively short history in Analysis. The scalar-valued theory was pioneered by the Swedish mathematician Gunnar Aronsson in the 1960s and since then has developed enormously. The general vector-valued case delayed a lot to be developed and its systematic development began in the 2010s. One of the most fundamental problems in the area which was completely open until today (and has been attempted by many researchers) concerned that of the title. In this talk I will discuss the first result in this direction.

6 dicembre

14:30, AULA 7

Nicola Garofalo (Università di Padova)

Sobolev and isoperimetric inequalities for Kolmogorov-Fokker-Planck operators

Abstract: In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order hypoellipitc evolution equation which displays many challenging features. Thirty years later, in the opening of his famous 1967 hypoellipticity paper, Hormander discussed a general class of degenerate Ornstein-Uhlenbeck operators that includes Kolmogorov's as a special case. The natural semigroups attached to such equations need not be symmetric or doubling, thus the existing theories do not readily apply. As a consequence, despite the large amount of work done by many people over the past thirty years, some basic questions have remained unsettled, such as Hardy-Littlewood-Sobolev and Isoperimetric inequalities and the study of local and nonlocal minimal surfaces. In this lecture I will present some interesting developments in the above program. This is joint work with Giulio Tralli.

22 novembre

11:00-17:30, a Pisa

MathAnalysis@UniFIPISI

Incontri di Analisi Matematica tra Firenze, Pisa e Siena

Organizzato da G. Bellettini, F. Bucci, E. Chiodaroli, A. Colesanti, E. Paolini, N. Visciglia.
La registrazione, il programma dettagliato ed ulteriori informazioni si trovano qui

15 novembre

14:30

Richard J. Gardner (Western Washington University)

The Minkowski Problem

Abstract

8 novembre

14:30

Vít Musil (Università degli Studi di Firenze e Czech Academy of Sciences)

Moser inequalities in Gauss space

Abstract

25 ottobre

14:30 e 15:30, AULA 7

Virginia Agostiniani (Università di Verona)

Symmetries for the torsion problem in a multiply connected domain

Luca Nenna (Université Paris Sud-Orsay)

Grand Canonical Optimal Transport

Abstract Agostiniani: In this talk, we discuss some recent symmetry results for the torsion problem, when the domain considered has multiple boundary components. This is a joint project with S. Borghini and L. Mazzieri.

Abstract Nenna: In this talk I will firstly review standard multi-marginal Optimal Transport (where a number N of marginals is fixed) focusing, in particular, on the applications in Quantum Mechanics (in this case the marginals are all the same and represent the electrons of an atom or a molecule). I will then extend the Optimal Transportation problem to the grand canonical setting: only an average number of marginals is now given (i.e. we can now define a OT problem with a fractional number of marginals), discussing existence, duality and numerical methods. I will finally compare these two problems and show how they behave differently despite considering the same cost functions. This a joint work with S. Di Marino and M. Lewin.

4 ottobre

14:30

Adolfo Arroyo Rabasa (University of Warwick)

Geometric rigidity of tangent symmetric gradient measures

Abstract

23 settembre 2019

14:30 e 15:30

Gamze Duzgun (Hacettepe University Ankara)

Existence of Three Nontrivial Solutions for a Nonlinear Fractional Laplacian Problem

Hacer Ilhan (Hacettepe University Ankara)

Similarity Searches in Motion Capture Databases

Organizzatori: Chiara Bianchini, Giuliano Lazzaroni.