Embedded Files
Simone Verzellesi (Università di Trento)
Ruled hypersurfaces in higher dimensional Heisenberg groups
12 luglio 2023 alle ore 11:30: in aula 2AB40
A typical issue in GMT is the characterization of global minimal hypersurfaces, that is the so-called Bernstein problem. While this topic is well understood in the Euclidean framework, the Bernstein problem in sub-Riemannian Heisenberg groups Hn leaves many interesting questions still unanswered. A crucial step in the study of minimal surfaces in H1 is to show that they are ruled by horizontal lines in a suitable sense. In this seminar, after a brief survey of the known results, we introduce a possible generalization of the notion of ruled surface to higher dimensional Heisenberg groups. After discussing some properties, we show some rigidity results for this condition, first in the class of non-characteristic hypersurfaces and then in the class of conical hypersurfaces. Finally, we make some remarks in order to understand the relationship between minimal hypersurfaces and ruled hypersurfaces.
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Carlo Gasparetto (Università di Pisa)
A short proof of Allard’s theorem
10 Luglio 2023 alle ore 11:30: in aula 2AB40
Allard’s theorem states that a minimal surface that is close enough to a plane coincides with the graph of a smooth function which enjoys suitable a-priori estimates. In this talk I show how to prove this result by exploiting viscosity techniques and a weighted monotonicity formula. Based on a joint work with G. De Philippis and F. Schulze.
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Cristian Mendico (Università di Roma Tor Vergata)
Non-coercive ergodic Hamilton-Jacobi-Bellman equations: controllable v.s. non-controllable case
4 Luglio 2023 alle ore 11:30: in aula 2AB40
The long-time average behavior of the value function in the classical calculus of variation is known to be connected with the existence of solutions of the so-called critical equations, that is, a stationary Hamilton-Jacobi equation which includes a sort of nonlinear eigenvalue called the critical constant (or effective Hamiltonian). In this talks, we will address a similar issues for the dynamic programming equation of an optimal control problem, with nonlinear state equation, for which coercivity of the Hamiltonian is non longer true. We will focus the attention on three particular cases, namely:
(i) control affine systems,
(ii) control of acceleration,
(iii) non controllable system with general state equation,
for which we can prove either the time-average asymptotic behavior or the complete long-time behavior. Besides the interests in all the positive answer we have, we will describe the open problems and the possible ways to solve them from the new ideas developed in the above cases. This talk is based on a series of research papers in collaboration with Piermarco Cannarsa, Pierre Cardaliaguet, Stephane Gaubert and Marc Quincampoix.
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Corentin Lena (Università di Padova)
Pleijel-type theorems for eigenfunctions
29 giugno 2023 alle ore 11:30: in aula 2AB40
Eigenvalue problems are often used to model the stationary states of physical systems. For instance, eigenfunctions of the Laplacian in a planar domain, with a Dirichlet boundary condition, describe the small oscillations of a membrane whose boundary is fixed: the membrane remains motionless where the eigenfunction vanishes. A natural question has emerged while studying this topic: in how many subdomains does the zero-set of an eigenfunction divide the original domain?
I will recall some tools used to tackle this problem and some classical results, focusing on the asymptotic upper bound discovered by Pleijel in 1956. I will then describe more recent extensions of Pleijel's theorem and in particular present joint work with Philippe Charron on Schrödinger operators.
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Federica Dragoni (Cardiff University)
Horizontal Mean Curvature flow: a Riemannian approximation
15 giugno 2023 alle ore 14:30: in aula 2AB40
In this talk I will give a brief overview on the evolution by horizontal mean curvature flow (i.e. the evolution in sub-Riemannian geometries) and compare that with the corresponding Euclidean and Riemannian evolutions. I will briefly introduce two methods for proving existence: the stochastic representation formula and the Riemannian approximation, showing then some connections between the two methods. The results are from a series of works, in collaboration with Nicolas Dirr, Raffaele Grande and Max von Renesse.
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Alberto Bressan (Penn State University)
Generic properties of first order Mean Field Games
14 giugno 2023 alle ore 14:30: in aula 2AB40
In this talk I shall briefly review some basic tools from differential geometry, that allow to prove generic results. As a first illustration of these techniques, uniqueness of solutions will be proved for a class of optimal control problems. Similar ideas can be applied to first order mean field games, in connection with existence, multiplicity, stability, and structural stability of solutions. Some results and more open problems will be discussed.
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Alessandro Goffi (Università di Padova)
Evans-Krylov estimates for fully nonlinear equations under weak concavity assumptions
5 giugno 2023 alle ore 14:30: in aula 2AB40
In 1982 L.C. Evans and N.V. Krylov proved interior a priori second derivatives estimates in Hölder spaces for fully nonlinear second order uniformly elliptic equations under the main assumption that the operator is either concave or convex in the Hessian variable. Since then, a remarkable question in the theory is to determine which hypotheses on the operator in between convexity/concavity and no assumptions ensure that solutions to general (Bellman-Isaacs) second order equations are classical. In this direction, N. Nadirashvili and S. Vladut exhibited counterexamples in dimension higher than or equal to 5 to show that the sole uniform ellipticity is in general not enough to reach classical regularity. Despite these developments, the minimal assumptions guaranteeing classical regularity are unknown, and the above question has remained largely open.
After a brief review of the regularity theory for fully nonlinear equations, in this talk I will discuss some improvements in the context of classical regularity and show how to prove interior C^{2,alpha} and C^{1,1} regularity for elliptic and parabolic problems under the assumption that the operator is either quasi-concave/convex (i.e. the superlevel/sublevel sets of F are convex). This improves upon a work by B. Andrews and lead, as a byproduct, to Schauder and Calderon-Zygmund estimates (above the Escauriaza exponent) under the same assumption on the nonlinearity through a perturbation argument by L. Caffarelli. Some further extensions to semiconcave operators and for functionals that are concave/convex or “close to a hyperplane” at infinity will also be discussed. The approach is based on linearization arguments and on Bernstein methods. Finally, I will show how the results imply polynomial Liouville theorems for solutions to elliptic problems on the whole space as well as for ancient solutions to parabolic equations.
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Fabio Cavalletti (SISSA)
Optimal Transport between algebraic hypersurfaces
30 maggio 2023 alle ore 12:30: in aula 2BC30
What is the optimal way to deform a projective hypersurface into another one? We will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic projective hypersurfaces. First, a natural topological embedding of the space of hypersurfaces of a given degree into the space of measures on the projective space is constructed. Then, the optimal transport problem between hypersurfaces is defined through a constrained dynamical formulation, minimizing the energy of absolutely continuous curves which lie on the image of this embedding. In this way an inner Wasserstein distance on the projective space of homogeneous polynomials is introduced. We will show the main properties of this distance and discuss applications on the regularity of the zeroes of a family of multivariate polynomials and on the condition number of polynomial systems solving.
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Giacomo Vianello (Università di Trento)
A vertex-skipping property for almost-minimizing sets in convex containers
23 maggio 2023 alle ore 12:30: in sala riunioni 701
Given a convex set K in the Euclidean 3-space, we focus on the boundary behavior of an almost-minimizer E for the relative perimeter in K. We show that the closure of the internal boundary of E cannot contain vertex-type singularities of the boundary of K. To prove this result, we exploit a blow-up argument that reduces the problem to the instability of a suitable plane passing through the vertex of a conical container. One of the intermediate results, that for instance allows us to consider a larger class of almost-minimizers, is a boundary monotonicity formula valid under some mild, extra assumptions on K.
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Benedetta Pellacci (Università della Campania Luigi Vanvitelli)
Spectral optimization problems in logistic heterogenous models
18 maggio 2023 alle ore 11:30: in presenza in aula 2AB40
The optimization of the principal eigenvalue of indefinite weighted problems settled in bounded domains arises as a natural task in the study of the survival thresholds in population dynamics. The study of the minimization of such eigenvalue with respect to the weight in the case of homogenenous Neumann, Dirichlet or Robin boundary conditions leads to a shape optimization problem completely solved only in dimension one. We will be focused on Neumann boundary conditions, performing some asymptotic analysis, corresponding to the case of either a complete hostile unfavourable region, or to an arbitrary small favourable one. We will study the optimizers, obtaining some qualitative properties, which, though widely expected, are still unknown in the general case. Some recent result for anisotropic diffusion will be also addressed. This is a joint research with Dario Mazzoleni (Università di Pavia) and Gianmaria Verzini (Politecnico di Milano), Giovanni Pisante (Università della Campania "Luigi Vanvitelli" and Delia Schiera (Istituto Superior Tecnico Lisbona).
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Giulio Galise (Roma Sapienza)
The generalized principal eigenvalue for nonlinear degenerate integral operators
21 aprile 2023 alle ore 11:30: in presenza in aula 2BC30
This talk is concerned with the validity of the Maximum Principle in bounded domains and its relation with the principal eigenvalue for a class of degenerate elliptic operators that are extremal among operators with one-dimensional fractional diffusion.
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Corrado Mascia (Roma Sapienza)
Turing instability for a class of hyperbolic reaction-diffusion systems
17 aprile 2023 alle ore 15:00: in presenza in aula 2BC30
Standard derivation of reaction-diffusion equations is based on two identities: balance of mass (which also involves the flux of the quantity under observation) and a corresponding definition of a constitutive identity for such flux, the standard choice for the former being the Fourier’s law. While very simple, such instantaneous response choice is not the unique selection. Following the approach of Extended Irreversible Thermodynamics, we consider a different law of relaxation type mechanism, possibly leading to a hyperbolic model. This talk focuses on the question of Turing instability, i.e. asymptotic stability with respect to homogeneous perturbations and instability with respect to spatial dependent perturbations. For such models, I will concentrate mainly on the case of two-dimensional systems, emphasizing the differences with respect to the usual parabolic modeling.
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Chiara Simeoni (Université Côte d'Azur)
Analysis and numerics of the propagation speed for hyperbolic reaction-diffusion models
17 aprile 2023 alle ore 14:00: in presenza in aula 2BC30
We briefly discuss different models for reaction-diffusion phenomena based on hyperbolic equations. The standard approach makes use of parabolic systems which are, indeed, well suited to explain events such as heat transmission in close-to-equilibrium regimes. Nevertheless, such modeling is criticizable for several reasons, for instance the prediction of an infinite speed of propagation, the lack of time-delay and related inertial effects and the exceptionality of well-posed boundary value problems. In addition, in many contexts the hyperbolic corrections are relevant for applications: dynamics of biological tissues, populations growth, forest fire models, etc... We adopt the point of view that a description by means of hyperbolic models - starting from the basic example o the telegraph equation - is viable, and more appropriate when the relaxation time required to perceive changes of the overall phenomenon is sufficiently large as compared to the diffusivity coefficient. As a matter of fact, differences may emerge in the transient regimes, whose cumulation may influence significantly the final outcome. Actually, the emphasis is placed on the numerical computation of the propagation speed of special traveling wave solutions, namely propagating fronts, of hyperbolic dissipative processes. Three basic numerical schemes are presented, two of which can also be applied to general hyperbolic systems (with reduced performance when dealing with discontinuous initial data), and we compare their performance with respect to providing effective approximations of the propagation speed. Therefore, we focus on a specific class of 2x2 systems corresponding to second order PDEs in one space dimension, which are adapted for simplified modeling of reaction-diffusion equations with monostable and bistable reaction terms. Beside the phase-plane algorithm which is convenient for approximating hyperbolic reaction-diffusion systems with damping, especially in cases with available explicit formulas, we propose two PDE-based numerical schemes, the so-called scout&spot algorithm - based on tracking the level curve of some intermediate value of the wave profile - and the LeVeque-Yee formula - given by the average value of the discrete transport velocity - by assessing their capability in comparative experiments. We draw attention to the fact that we do not merely regard hyperbolic equations as perturbation of the limiting parabolic counterparts, and then the corresponding numerical schemes as a tool for approximating parabolic equations. But rather, we focus on hyperbolic models considered as an alternative language for describing dissipative mechanisms which are particularly interesting in far-from-equilibrium regimes.
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Pierpaolo Soravia (Universita` di Padova)
Higher order hamiltonians for small time attainability of manifolds
14 aprile 2023 alle ore 11:30: in presenza in aula 2AB40
We will discuss a new approach to small time local attainability of smooth manifolds of any dimension, possibly with boundary and to prove Holder continuity of the minimum time function. We give explicit pointwise conditions of any order by using higher order hamiltonians which combine derivatives of the controlled vector field and the functions that locally define the target. For the controllability of a point our sufficient conditions extend some classically known results for symmetric or control affine systems, using the Lie algebra instead. Our sufficient higher order conditions are explicit and easy to compute for targets with curvature and general control systems.
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Daniele Castorina (Università di Napoli Federico II)
Mean field sparse optimal control of systems with additive white noise
13 aprile 2023 alle ore 12:30 in presenza: in aula 1BC30
We analyze the problem of controlling a multi-agent system with additive white noise through parsimonious interventions on a selected subset of the agents (leaders).
For such a controlled system with a SDE constraint, we introduce a rigorous limit process towards an infinite dimensional optimal control problem constrained by the coupling of a system of ODE for the leaders with a McKean-Vlasov-type SDE, governing the dynamics of the prototypical follower. The latter is, under some assumptions on the distribution of the initial data, equivalent with a (nonlinear parabolic) PDE-ODE system.
The derivation of the limit mean-field optimal control problem is achieved by linking the mean-field limit of the governing equations together with the $\Gamma$-limit of the cost functionals for the finite dimensional problems. Joint work with Giacomo Ascione (SSM Napoli) and Francesco Solombrino (Napoli Federico II).
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Alessandra Pluda (Università di Pisa)
Steiner Problem, global and local minimizers of the length functional
23 marzo 2023 alle ore 12:00: in presenza in aula 2AB40
The Steiner problem, in its classical formulation, is to find the 1-dimensional connected set in the plane with minimal length that contains a finite collection of points. Although existence and regularity of minimizers is well known, in general finding explicitly a solution is extremely challenging, even numerically. A possible tool to validate the minimality of a certain candidate is the notion of calibrations. In this talk I will introduce the different definitions of calibrations for the Steiner problem available in the literature, I will give example of existence and non—existence of calibrations and I will show how one can easily get informations on both global and local minimizers.
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Yoshihiro Tonegawa (Tokyo Institute of Technology)
Existence of generalized solution of mean curvature flow
20 marzo 2023 alle ore 12:30 in presenza: in aula 1BC45
I present a recent time-global existence result of mean curvature flow (MCF) in the setting of geometric measure theory. The new solution is a MCF in two different weak formulations: it is a Brakke flow and a generalized BV solution, the latter being slightly different from original BV solution proposed by Luckhaus-Struzenhecker. I plan to spend most time explaining about the notion and result, and give a brief outline of the proof. This is a joint work with Salvatore Stuvard and is also based heavily on a previous work with Lami Kim.
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Alberto Maione (University of Freiburg)
Variational methods for a class of mixed local–nonlocal operators
13 marzo 2023 alle ore 14:30 in presenza: in aula 2AB40
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Charles Bertucci (École polytechnique)
Mean Field Games with incomplete information
06 marzo 2023 alle ore 14:30 in presenza: in aula 2AB40
In this talk, I will revisit the classical theory of MFG in a context in which the players cannot observe the states of the other players. I will first study a case in which the players are totally blind, i.e. they will not gain any information. I shall then present the more involved case in which the players observe their payments, and thus deduce some information on the repartition of players.
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Eugenio Pozzoli (Universita` di Padova)
Bilinear controllability of wave equations in small time
27 febbraio 2023 alle ore 14:30 in presenza: in aula 2AB40
The controllability of evolution equations, that is the capability of steering any initial state to any final state by suitably acting on the system with a control, has important applications all across natural sciences. In order to obtain time-optimal results, it is natural to ask which states can be reached in very short times. In bilinear control, the problem of small time controllability is open for many PDEs. In this talk we will focus on the case of wave equations, presenting a small time controllability result and the ideas (of geometric control flavour) behind its proof.
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Thierry Paul (LJLL & CNRS, Sorbonne University)
From microscopic to macroscopic: large number dynamics of multi-agent and cells, possibly interacting with a chemical background
05 dicembre 2022 alle ore 11:30 in presenza: in aula 2AB45
We will present several results obtained with Marta Menci (Campus Bio-Medico, Roma), Roberto Natalini (IAC-Cnr Roma) and Emmaneul Trélat (Sorbonne Université, Paris) concerned with the collective motion involving N agents (possibly distinguishable) or N cells subjects to chemotaxis as N becomes large. The goal will be to describe all the different steps forming the way along which genuine non-linear partial differential equations posed on the one particle-space (Vlasov, Euler, graph-limit) can be derived out of the ordinary differential equations driving the microscopic dynamics. Recent numerical simulations will be presented, showing how the striking effects, e.g. of alignment dynamics, remain present all the way in the transition, including the macroscopic scale. Finally we will show that "any" quasi-linear PDE can be seen as deriving from a multi-agent system on the limit of large numbers of agents.
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Lauro Silini (ETH Zürich)
The isoperimetric conjecture in the complex hyperbolic space: Hopf-symmetries and densities
24 novembre 2022 alle ore 11:30 in presenza: in aula 2AB40
Rank one symmetric spaces of non-compact type are completely classified to be the real, complex and quaternionic hyperbolic spaces and the (octonionic) Cayley plane. Their rich isometry group, together with the classical symmetrization procedures introduced by Steiner and Schwarz, lead to the natural conjecture formulated by Gromov and Ros that in this context geodesic balls are isoperimetric regions. Up to now, this is known to be true only in the real hyperbolic setting. In this seminar, we will give a new positive answer restricting our attention to a class of sets sharing a particular radial symmetry.
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Alessandro Alla (Universita` Ca' Foscari di Venezia)
Recent algorithms towards the control of PDEs via dynamic programming equations
21 novembre 2022 alle ore 11:30 in presenza: in aula 2BC30
Nonlinear feedback laws can be computed via the value function characterized as the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation which stems from the dynamic programming approach. Semi-lagrangian schemes for the discretization of the dynamic programming principle are based on a time discretization projected on a state-space grid. The use of a structured grid makes this approach not feasible for high-dimensional problems due to the curse of dimensionality.
In this talk, I will present two recent algorithms that mitigate the curse of dimensionality based on meshes driven by the dynamics of the problem studied. In the first part of the talk, I will introduce a tree-structure algorithm for finite horizon control problems whereas, in the second part, the use of Radial basis functions for infinite horizon control problems. Error estimates and numerical experiments will be provided throughout the talk to show the effectiveness of methods discussed.
Joint works with M. Falcone, H. Oliveira, L. Saluzzi and G. Santin.
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Philippe Souplet (Université Sorbonne Paris Nord)
Diffusive Hamilton-Jacobi equations and their singularities
14 novembre 2022 alle ore 11h30 in presenza: in aula 2AB45
We consider the diffusive Hamilton-Jacobi equation $u_t-\Delta u=|\nabla u|^p$ with homogeneous Dirichlet boundary conditions, which plays an important role in stochastic optimal control theory and in certain models of surface growth (KPZ). Despite its simplicity, in the superquadratic case p>2 it displays a variety of interesting and surprising behaviors. We will discuss two classes of phenomena:
- Gradient blow-up (GBU) on the boundary: time rate, single-point GBU, space and time-space profiles, Liouville type theorems and their applications;
- Continuation after GBU as a global viscosity solution with loss and recovery of boundary conditions.
In particular, we will present the recently obtained, complete classification of solutions in one space dimension, which describes the losses and recoveries of boundary conditions at multiple times, as well as all the possible GBU and recovery rates. This talk is based on a series of joint works in collaboration with A. Attouchi, R. Filippucci, Y. Li, N. Mizoguchi, A. Porretta, P. Pucci, Q. Zhang.
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Vincenzo Vespri (Università di Firenze)
Harnack estimates for anisotropic operators
2 novembre 2022 alle ore 12h15 in presenza: in aula 2BC30
The theory of anisotropic operators is fragmented. Only few regularity results are available when the coefficients are not regular. In this talk we will speak of the state of the art and of the open questions.
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Martino Bardi (Università di Padova)
An Eikonal equation with vanishing Lagrangian arising in Global Optimization.
24 ottobre 2022 alle ore 11h30 in presenza: in aula 2AB40
Global unconstrained optimization of a non-convex function f in Euclidean space is a classical problem that is attracting considerable new attention in view of applications to the training of deep neural networks, especially for highly nonlinear, possibly non-smooth loss functions f in very high space dimension.
We show a connection between this problem and a "critical" eikonal-type equation arising also in ergodic control and weak KAM theory. A solution v of the critical Hamilton-Jacobi equation is built by a small discount approximation as well as the long time limit of an associated finite horizon optimal control problem. Then v is represented as the value function of a control problem with target, whose optimal trajectories are driven by a differential inclusion describing the gradient descent of v. Such trajectories, i.e. the gradient orbits of v, are proved to converge to the set of minima of f.
Finally we discuss whether the optimal trajectories reach the minima of f in finite time. This is related to some classical problems on gradient dynamical systems, that we briefly recall and compare with our result. This is joint work with Hicham Kouhkouh (arXiv:2202.02561).
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