Seminari anno accademico 2021-2022

Guido de Philippis (Courant Institute)

Non-degenerate minimal surfaces as energy concentration sets: a variational approach

Giovedì 21 luglio 2022 alle ore 11h30 in modalità duale: in aula 2BC30 e via Zoom

I will show that every non-degenerate minimal sub-manifold of codimension 2 can be obtained as the energy concentration set of a family of critical points of the (rescaled) Ginzburg Landau functional. The proof is purely variational, and follows the strategy laid by Jerrard and Sternberg in 2009. The same proof applies also to the Yang-Mills-Higgs and to the Allen-Cahn-Hillard energies. This is a joint work with Alessandro Pigati.

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Frank Duzaar (University of Erlangen)

Higher integrability for doubly nonlinear parabolic equations

Martedì 19 luglio 2022 alle ore 11h30 in modalità duale: in aula 1BC50 e via Zoom

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Diogo Gomes (KAUST)

Monotone operator methods in mean-field games

Martedì 5 luglio 2022 alle ore 14:30 in modalità duale: in aula 1BC45 e via Zoom

Mean-field games model systems with a large number of competing rational agents. Often these games are determined by a system of a Hamilton-Jacobi equation coupled with a transport or Fokker-Planck equation. In many cases, this system can be regarded as a monotone operator. This structure is at the basis of the uniqueness proof by Lasry and Lions. Using monotone operator techniques, we show how to prove the existence of solutions in a number of cases, establish uniqueness and show how they can be used to construct numerical methods.

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Daniele Valtorta (University of Zurich)

Singularities for Q-valued Dirichlet minimizing functions

Martedi` 21 giugno 2022 alle ore 14:15 in modalità duale: in aula 1BC45 e via Zoom

We present new regularity results for the singular sets of Q-valued harmonic functions. This work is a collaboration with Camillo de Lellis, Andrea Marchese and Emanuele Spadaro.

For a Q-valued Dirichlet minimizing harmonic functions, we show that its singular set is m-2 rectifiable. Moreover, we also prove uniform Minkowski bounds for the set of Q-points. The proof is based on a Reifenberg-type theorem obtained in collaboration with Aaron Naber, and the technique is very versatile in nature and can be used to tackle many different problems in GMT. For example, this technique can be adapted to study the singularities of thin obstacle problems and liquid crystals.

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Paolo Luzzini (Università di Padova)

Asymptotics of buckling eigenvalues

Giovedì 9 giugno 2022 alle ore 11:30 in modalità duale: in aula 2BC30 e via Zoom.

Since the seminal works of Hermann Weyl at the beginning of the 20th century, several authors have investigated the spectral asymptotics of partial differential operators. Following this tradition, in this talk I will first present a recent result on a new proof of Weyl’s law for the buckling eigenvalues requiring minimal assumptions on the domain. The proof relies on asymptotically sharp lower and upper bounds that we develop for Riesz means. Moreover, we compute the second term in Weyl’s law in the case of balls and bounded intervals. This, together with some formal considerations, leads us to state a conjecture for the second term in general domains.

The talk is based on a joint work with Davide Buoso (UPO), Luigi Provenzano (Sapienza Universita` di Roma), and Joachim Stubbe (EPFL).

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Elia Brué (IAS Princeton)

The metric measure boundary of spaces with Ricci curvature bounded below

Giovedì 19 maggio 2022 alle ore 15:00 tramite Zoom

Kapovitch, Lytchak, and Petrunin introduced the notion of metric measure boundary in an attempt to solve a long-standing conjecture of Perelman-Petrunin about the existence of infinite geodesics on Alexandrov spaces. They proved that the conjecture is satisfied provided the metric measure boundary of any Alexandrov space without boundary is vanishing. However, the latter turned out to be a difficult question and remained open. In this talk, I will present a recent paper in collaboration with A. Mondino and D. Semola, where we solve the Perelman-Petrunin conjecture by proving that the metric measure boundary is vanishing on any RCD space without boundary.

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Marta Strani (Università Ca' Foscari Venezia)

Fast-slow dynamics in evolutive PDEs with nonlinear diffusions

Mercoledì 11 maggio 2022 alle ore 11h30 in modalità duale: in aula 2AB40 e via Zoom.

We present some results on the long-time dynamics of the solutions to different evolutive PDEs, which present both convective and/or reaction terms, together with nonlinear diffusions. In particular, we consider models with monotone bounded and unbounded diffusions (as, for instance, the mean curvature operator), as well as one with the p-Laplacian operator. After proving the existence of stable steady states for the systems under consideration, we study the asymptotic dynamics of the time dependent solutions, showing their slow motions towards the equilibrium configuration.

These results are obtained in collaboration with R. G. Plaza and R. Folino, Institute of Applied Mathematics and Systems (IIMAS), National Autonomous University of Mexico (UNAM)

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Luigi De Masi (SISSA)

Min-max theory for minimal surfaces

Giovedì 28 Aprile 2022 alle ore 15:30 in modalità duale: in aula 2BC60 e via Zoom.

The most famous context where minimal surfaces, i.e. stationary points for area functional, arise is the Plateau Problem: one tries to minimize the area of a surface whose boundary is a fixed closed curve.

With different topological/boundary constraints, minimizing methods may provide just trivial solutions. In that cases one has to rely on different ways to obtain a non-trivial critical point for a suitable functional. Min-max methods have been successfully used for many of these problems.

In this talk I will explain the basic ideas of min-max theory applied to the problem of finding a minimal surface $\Sigma$ in a container $\mathcal{M} \subset \mathbb{R}^3$ with a fixed angle condition at the boundary. This is based on a joint work with Guido De Philippis.

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Giulio Tralli (Università di Padova)

Fundamental solutions and groups of Heisenberg type

Giovedì 7 aprile 2022 alle ore 11:30 in modalità duale: in aula 2BC30 e via Zoom.

In this talk we discuss the conformal fractional powers of the horizontal Laplacian in Heisenberg-type groups. We present a new approach, based on the heat kernels of suitable extension operators, to the derivation of the fundamental solutions for these nonlocal operators and to the explicit construction of the Aubin-Talenti type functions. Frequent comparisons with the Euclidean case of the powers of the Laplacian will be made, focusing both on the similarities of the intertwining properties and on the main differences between the underlying heat kernels. The talk is based on a joint project with N. Garofalo.

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Marco Michetti (Université de Lorraine)

A comparison between Neumann and Steklov eigenvalues

Martedì 22 febbraio 2022 alle ore 11:30 in modalità duale: in aula 2BC30 e via Zoom.

Abstract: In this talk we present a comparison between the normalized first (non-trivial) Neumann eigenvalue |Ω|μ_1(Ω) for a Lipschitz open set Ω in the plane, and the normalized first (non-trivial) Steklov eigenvalue P(Ω)σ_1(Ω). More precisely, we study the ratio F(Ω) := |Ω|μ_1(Ω)/P(Ω)σ_1(Ω). We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets Ω. Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds.In the last part of the talk we present the corresponding Blaschke-Santalo diagrams (x, y) =(|Ω|μ_1(Ω), P(Ω)σ_1(Ω)) and we state some open problems.This talk is based on a joint work with Antoine Henrot.

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Davide Barilari (Universita` di Padova)

Control systems and comparison theorems in geometry

Mercoledì 9 febbraio 2022 alle ore 11:30 in modalità duale: in aula 2AB40 e via Zoom.

Abstract: In this talk we want to discuss how (and in which sense) basic models for control systems (linear quadratic control systems) can be used as model spaces in geometry, in particular when considering comparison-type theorems.

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Luciano Marzufero (Universita` di Trento)

A hybrid control framework for an optimal visiting problem: from a single player to a crowd

Mercoledi` 26 gennaio 2022 alle ore 11:30 tramite Zoom

Abstract: In an optimal visiting problem, the aim is to control a trajectory that has to touch or pass as close as possible to a collection of target points or regions. We introduce a hybrid control-based approach for the classic problem where the trajectory can switch between a group of discrete states related to the targets of the problem. The model is subsequently adapted to a mean-field game framework to study viability and crowd fluxes to model a multitude of indistinguishable players.

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Daria Ghilli (LUISS)

Time-dependent focusing Mean Field Games with strong aggregation

Martedì 18 gennaio 2022 alle ore 11:30 in modalità duale: in aula 1BC45 e via Zoom.

Abstract: Mean Field Games (MFG) study Nash equilibria in (differential) games with a population of infinitely many identical players. In this talk we are interested in the ”focusing” case, that is, MFG where the coupling has a decreasing character in m, m being the density of the population. Since the coupling models the cost of a single agent in terms of m, these systems model situations in which agents aim at maximizing aggregation. Differently from the case where the coupling is monotone increasing (to which large part of the literature is devoted), phenomena such as non existence, non uniqueness and periodic solutions may appear. We will analyze the model case in which the coupling is −σm^α, α ≥ 2/N, where N is the dimension of the state space. We will show existence for σ small, and non existence for σ large whenever the time horizon T is large.

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Alberto Maione (Albert-Ludwigs-Universität Freiburg)

Variational convergences for functionals and differential operators depending on vector fields

Giovedì 13 gennaio 2022 alle ore 11:30 in modalità duale: in aula 2AB40 e via Zoom.

In this talk, I present results concerning variational convergences for functionals and differential operators depending on a family of Lipschitz continuous vector fields X. This setting was introduced by Folland and Stein and has recently found numerous applications in the literature. The convergences taken into account date back to the 70’s and are Γ-convergence, introduced by Ennio De Giorgi and Tullio Franzoni, dealing with functions and functionals, and H-convergence, whose theory was initiated by François Murat and Luc Tartar and which deals with differential operators. The main result presented today is a Γ-compactness theorem which ensures that sequences of integral functionals depending on the family X, with standard regularity and growth conditions, Γ-converge in the strong topology of Lp (p > 1) to a functional belonging to the same class, by assuming an algebraic condition on X. As an interesting application of the Γ-compactness theorem, I finally show that the class of linear differential operators in X-divergence form is closed in the topology of the H-convergence. The variational technique adopted to this aim relies on a new approach recently introduced by Nadia Ansini, Gianni Dal Maso and Caterina Zeppieri.

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Paolo Gidoni (UTIA, Czech Academy of Sciences, Praga)

A mathematical perspective on the quasi-static approximation for crawling locomotion

Lunedi` 20 dicembre 2021 alle ore 12:30 in modalità duale: in aula 2AB40 e via Zoom.

Abstract: The quasi-static limit is a convenient approximation in the modelling of several suitable mechanical systems, when the evolution occurs at a sufficiently slow time-scale. One of the situations where the quasi-static approximation is usually adopted is the study of locomotion in earthworms, snails and other limbless terrestrial animals, as well as in an increasing number of soft-robotic devices mimicking such strategies. In this seminar, we discuss the meaning of the quasi-static assumption in soft-bodied crawlers by a mathematical perspective. First, we survey the relevance, or lack thereof, of inertia in some locomotion strategies. Then, we focus on a basic, but effective, family of models for soft crawlers and provide a mathematically rigorous derivation of quasi-static limit. More precisely, we show the uniform convergence of dynamical solutions to a (quasi-static) rate-independent system, in a slow-actuation limit.

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Cristian Mendico (Università di Padova)

Asymptotic behavior of solutions to Hamilton-Jacobi-Bellmann equations

Venerdì 10 dicembre 2021 alle ore 11:30 in modalità duale: in aula 2AB45 e via Zoom.

Abstract: The analysis of the ergodic behavior of solutions to Hamilton-Jacobi-Bellmann equations has a long history going back to the seminal paper by [Lions, P.-L., Papanicolaou, G. and Varadhan, S.R.S]. Since this work, the subject has grown very fast and when the Hamiltonian is of Tonelli type a large number of results have been proved. However, few results are available if the Hamiltonian fails to be Tonelli, i.e., the Hamiltonian is neither strictly convex nor coercive with respect to the momentum variable. In particular, such results cover only some specific structure and so, the general problem is still open. In this talk, I will present some recent results obtained in collaboration with Piermarco Cannarsa and Pierre Cardaliaguet concerning the long time-average behavior of solutions to Hamilton-Jacobi-Bellman equations arising from optimal control problems with control of acceleration, first, and then from optimal control problems of sub-Riemannian type. We will show the existence of a critical constant in both cases but the existence of a critical solution only in case of sub-Riemannian geometry. For this latter case, we also show some results on the Aubry set. We conclude presenting open problems and ideas toward the solution of the general case.

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Raffaele Grande (Cardiff University, UK)

Stochastic control problems and evolution by horizontal mean curvature flow

Lunedi` 15 novembre 2021 alle ore 12:30 tramite ZOOM

Abstract: The horizontal mean curvature flow is widely used in neurogeometry (e.g. Citti-Sarti model for the visual cortex) and in image processing. It represents the contracting evolution of a hypersurface embedded in a particular geometrical setting, called sub-Riemannian geometry, in which only some 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. To avoid this problem, it is possible to use the notion of Riemannian approximation applied to the horizontal mean curvature flow.

In this talk I will explain 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 Riemannian case and Dirr, Dragoni and von Renessee in the sub-Riemannian setting). At the end, 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) and about the convergence of approximated solutions to the horizontal solutions.

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Richard Vinter (Imperial College)

The Euler Lagrange Condition of Optimal Control: Historical Perspective and Recent Improvements

venerdì 24 settembre 2021 alle ore 12:00 in modalità duale: in aula 2AB40 e via Zoom.

Abstract: First order necessary conditions for optimal control problems in which the dynamic constraint is modelled as a differential inclusion have been known for many years. A definitive version is provided by Clarke's 2005 Memoir. The key ingredients are the generalized Euler Lagrange inclusion (replacing the costate equation of classical optimal control), the transversality condition and the Weierstrass (or Hamiltonian maximization property) condition. Clarke and de Pinho's 2010 paper provided an important application of this theory, using it as a starting point to derive necessary conditions satisfied by minimizers for problems, having a "controlled differential equation" formulation and involving mixed state/control constraints.

In his 2019 paper, Ioffe added a refinement to the Weierstrass condition, identifying a larger set of controls over which the Hamiltonian must be maximized. We shall explore, through new theory and examples, the significance of this refinement. We derive new necessary conditions for mixed constraint problems involving controlled differential equations, via a reduction to a differential inclusion problem, that, for the first time, incorporate Ioffe's refinement. Two examples, concerning differential inclusion problems and controlled differential equations problems with mixed constraints, are presented that show how the extra tests present in the refined Weierstrass conditions can be used to identify extremals.