## Seminario di Analisi Matematica

SEMINARI A.A. 2021/2022

22/11/2021

Nicola Garofalo
Heat kernels for a class of evolution equations of hybrid type

The aim of my talk is to construct (explicit) heat kernels for some evolution equations which arise in physics, conformal geometry and subelliptic PDEs. A common feature is that the relevant partial differential operator appears in the form L_1 + L_2 - D_t, but the variables of L_1 and L_2 cannot be decoupled. This obstruction results in the fact that the heat kernel of L_1 + L_2 - D_t cannot be written as the product of the heat kernels of the operators L_1  - D_t and L_2 - D_t. One of the highlights will be the construction of the heat kernel for the time-dependent version of an extension problem" introduced by Frank, Del Mar Gonzalez, Monticelli and Tan. Such problem plays a pervasive role in conformal CR geometry and, among other things, it serves as a way to define the fractional powers of the conformal sublaplacian on the Heisenberg group. The leitmotif of my talk is emphasising the so far unexplored connection of the relevant hybrid equations with the heat kernel of the generalised operator of Ornstein-Uhlenbeck type in the opening of H\"ormander's groundbreaking 1967 work on hypoellipticity. This is joint wok with Giulio Tralli.

29/11/2021

Paolo Piccione
Nonplanar minimal spheres in ellipsoids of revolution

In 1987, Yau posed the question of whether all minimal 2-spheres in a 3-dimensional ellipsoid inside R^4 are planar, i.e., determined by the intersection with a hyperplane. While this is the case if the ellipsoid is nearly round, Haslhofer and Ketover have recently shown the existence of an embedded non-planar minimal 2-sphere in sufficiently elongated ellipsoids, with min-max methods. Using bifurcation theory and the symmetries that arise in the case where at least two semi-axes coincide, we show the existence of arbitrarily many distinct embedded non-planar minimal 2-spheres in sufficiently elongated ellipsoids of revolution. This is based on joint work with R. G. Bettiol (CUNY).

06/12/2021

Cristian Mendico
Asymptotic behavior of solutions to Hamilton-Jacobi-Bellmann equations

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-Papanicolaou-Varadhan. 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.

13/12/2021

Anna Abbatiello
On the existence of solutions for a generalized Navier-Stokes-Fourier system and their stability analysis

We consider flows of non-Newtonian heat conducting incompressible fluids in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field
and the spatially inhomogeneous Dirichlet boundary condition for the temperature. The ultimate goal is to show that the fluid converges to equilibrium as time tends to infinity.
However, to justify such result, one needs to deal with very special inequalities and very special test functions, which are typically not admissible on the level of weak solutions.
We show how one can overcome such difficulties. We define a notion of renormalized entropy solution and for large class of non-Newtonian fluids we establish its existence.
Such solutions are then perfectly prepared for proving the stability result. This is a joint work with M. Bulıcek and P. Kaplicky.

Università degli Studi di Roma Sapienza
17/01/2021

Luigi Carlo Berselli  ANNULLATO
Pseudo monotone operators and the unsteady rotational Smagorinsky model

We show that the rotational Smagorinsky model for turbulent flows can be put in the setting of Bochner pseudo-monotone evolution equations. This allows to prove existence of weak solutions identifying a proper weighted spaces and checking some easily verifiable assumptions, at fixed time. We also will briefly discuss the critical role of the exponents present in the model (power of the distance function and power of the curl).

Università di Pisa
24/01/2022

Eris Runa
Symmetry breaking and pattern formation for local/nonlocal interaction functionals

In this talk we will review some recent results we obtained on the one-dimensionality of the minimizers
of a family of continuous local/nonlocal interaction functionals in general dimension. Such functionals have a local term, typically the perimeter or its Modica-Mortola approximation, which penalizes interfaces, and a nonlocal term favouring oscillations which are high in frequency and in amplitude. The competition between the two terms is expected by experiments and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used  to model pattern formation, either in material science or in biology. The difficulty in proving the emergence of such structures is due to the fact that the functionals are symmetric with respect to permutation of coordinates, while in more than one space dimensions minimizers are one-dimensional, thus losing the symmetry property of the functionals. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnetic systems, respectively  colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers are one-dimensional and periodic, in general dimension and also while imposing a nontrivial volume constraint.

Deutsche Bank
31/01/2022

Giulio Ciraolo
Classification and nonexistence results for critical p-Laplace type equations

We present some sharp results for nonnegative solutions of nonlinear PDEs of $p$-Laplace type (possibly anisotropic). These PDEs are critical in the sense that they are associated with the study of critical points of functional inequalities. In this talk we consider critical equations arising from Sobolev and Caffarelli-Kohn-Nirenberg (CKN) inequalities.
We discuss classification results in convex cones and prove that the solutions have radial symmetry [1], non-existence results in bounded convex domains [2] and the occurrence of symmetry breaking for CKN inequalities [3].
[1] G. Ciraolo, A. Figalli, A. Roncoroni. Symmetry results for critical anisotropic p-Laplacian equations in convex cones. Geom. Funct. Anal., 30 (2020), 770-803.
[2] G. Ciraolo, R. Corso, A. Roncoroni. Classification and non-existence results for weak solutions to quasilinear elliptic equations with Neumann or Robin boundary conditions. J. Funct. Anal., 280 (2021), 108787.[3] G. Ciraolo, R. Corso. Symmetry for positive critical points of Caffarelli-Kohn-Nirenberg inequalities. Nonlinear Anal., 216 (2022), 112683.

Università degli Studi di Milano
07/02/2022

Alessandro Iacopetti
New regularity results for the prescribed mean curvature equation in the Lorentz-Minkowski space

In this talk we present some recent results concerning the existence and the regularity of weak solutions of the prescribed mean curvature equation in the Lorentz-Minkowski space (for spacelike hypersurfaces), when the mean curvature is in L^p.
In the first part of the talk we will show a new gradient estimate for entire smooth solutions of the prescribed mean curvature equation. Then we will prove that, if p>N, then the unique minimizer of the Born-Infeld energy, which is a priori only Lipschitz continuous, is actually a strictly spacelike weak solution of the equation and it is of class W^{2,p}_{loc}. Finally we will discuss some open problems.
These results are collected in a series of joint works with Prof. D. Bonheure (Université Libre de Bruxelles).

Università degli Studi di Torino

14/02/2022

Xavier Ros Oton
The singular set in the Stefan problem

The Stefan problem, dating back to the XIXth century, is probably the most classical and important free boundary problem. The regularity of free boundaries in the Stefan problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is $C^\infty$ in space and time, outside a certain set of singular points.
The fine understanding of singularities is of central importance in a number of areas related to nonlinear PDEs and Geometric Analysis. In particular, a major question in such a context is to establish estimates for the size of the singular set. The goal of this talk is to present some new results in this direction for the Stefan problem in $\R^3$. This is a joint work with A. Figalli and J. Serra.

ICREA & Univerisitat de Barcelona
21/02/2022

Lucia De Luca
Crystallization results for pairwise interaction energies in two dimensions

I will discuss some recent crystallization results for pairwise interaction energies of systems of particles in the plane. I will focus on the so-called Heitmann-Radin (HR) sticky disc potential that - in its classic form - is defined by V(r)=+infty for r<1, V(r)=-1 for r=1, V(r)=0 elsewhere. For the classic HR functional it has been proven that minimizing configurations are subsets of the regular triangular lattice. First, I will show how this result extends in a suitable sense to the class of quasi-minimizers. Furthermore, I will enrich the classic HR model in order to deal with vectorial crystallization problems arising in mathematical biology. Specifically, associating a vectorial orientation to each particle of the configuration and enforcing threshold criteria for the interactions between particles, I will show that minimizing configurations exhibit the so-called diamond formation (typical in fish schooling).

Istituto per le Applicazioni del Calcolo - CNR
28/02/2022 nessun seminario
07/03/2022 nessun seminario
14/03/2022

Gabriella Tarantello
A Donaldson functional for CMC-immersions of surfaces in Hyperbolic 3-space

We discuss CMC immersions of close surfaces (orientable and of genus larger than 1) in hyperbolic 3-spaces and their moduli spaces in the framework of Teichmuller theory. To this purpose we consider a Donaldson type functional introduced by Gonsalves-Uhlenbeck (2007) and establish uniqueness for the corresponding critical points. By this result we will be able to parametrise the moduli space of the immersions in terms of elements of the tangent bundle of the Teichmuller space.

Università degli Studi di Roma
Tor Vergata
21/03/2022

Riccardo Scala
On the relaxation of the area functional and the plateau problem

We discuss the setting of the nonparametric plateau problem and introduce the relaxation of the area functional. We present some recent developments on the analysis of it with the aid of cartesian currents.

Università di Siena
28/03/2022

Daniela Tonon
A comparison principle for Hamilton-Jacobi equations on infinite dimensional spaces

In this talk, we present a comparison principle for the Hamilton Jacobi (HJ) equation corresponding to linearly controlled gradient flows of an energy functional defined on a metric space. The main difficulties are given by the fact that the geometrical assumptions we require on the energy functional do not give any control on the growth of its gradient flow nor on its regularity. Therefore this framework is not covered by previous results on HJ equations on infinite dimensional spaces (whose study has been initiated in a series of papers by Crandall and Lions). Our proof of the comparison principle combines some rather classical ingredients, such as Ekeland’s perturbed optimization principle, with the use of the Tataru distance and of the regularizing properties of gradient flows in evolutional variational inequality formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian.  Our abstract results apply to a large class of examples, including gradient flows on Hilbert spaces and  Wasserstein spaces equipped with a displacement convex energy functional satisfying McCann’s condition.

04/04/2022

Konstantinos Zemas
Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces

Quantitative rigidity results, besides from their inherent geometric interest, have played a prominent role in the mathematical study of models related to elasticity\plasticity. For instance, the celebrated rigidity estimate of Friesecke, James, and Müller has been widely used in problems related to linearization, discrete-to-continuum or dimension-reduction issues for functionals within the framework of nonlinear elasticity.
In this talk, I will present a generalization (in the physically relevant dimensions d=2,3) of this result to the setting of variable domains, where the geometry of the domain comes into play, in terms of a suitable integral curvature functional of its boundary. The estimate can be used to establish compactness in the space of generalized special functions of bounded deformation for sequences of displacements related to deformations with uniformly bounded elastic energy.
As an application, we rigorously derive linearized models for nonlinearly elastic materials with free surfaces by means of Gamma-convergence. In particular, we study energies related to epitaxially strained crystalline films and to the formation of material voids inside elastically stressed solids.
This is joint work with Manuel Friedrich and Leonard Kreutz

### Universität Münster

11/04/2022

Flavia Smarrazzo
Radon measure-valued solutions for a class of evolution equations with degenerate coerciveness and measure data

Initial-boundary value problems for nonlinear parabolic equations ut = ∆ɸ having Radon measures as initial data have been widely investigated, looking for solutions which for positive times take values in some function space. More recently, solutions have been defined and investigated, which for positive times take values in the space of finite Radon measures. Such solutions are often called Radon measure-valued, to distinguish them from function-valued solutions. In general, whether the solution of the problem is a function for some time t>0 can be regarded as a M-L1 regularizing effect.

The aim of this talk is to address well-posedness and regularity results, depending on whether or not the initial data charge sets of suitable capacity (determined by the growth order of ɸ), and on suitable compatibility conditions, describing the behaviour of the singular part of solutions. The diffusion function ɸ is only assumed to be continuous, nondecreasing and at most powerlike: no assumptions about existence or estimates from below of the diffusivity ɸ’ are made (except for some regularization results). This lack of regularity and strong coercivity, along with the possible occurrence of infinitely many nondegenerate intervals where ɸ is constant, requires using more refined compactness arguments and Young measure techniques in the proof of existence, which is in turn obtained by a suitable approximation procedure of the initial measure. The possible occurrence or lack of instantaneous M-L1 regularizing effects, as well as partial uniqueness results, will also be discussed.

Università Campus Bio-Medico Roma
02/05/2022

Andrea Corli
Wavefront solutions to reaction-convection equations with Perona-Malik diffusion

This talk is concerned with traveling-wave solutions to nonlinear parabolic equations of forward-backward type. First, a short review of some results where the diffusion coefficient depends on the unknown function u (but not on its derivatives) is given. Then we focus on the case where the diffusion coefficient depends on the spatial derivative of u. A famous example is Perona-Malik’s equation in image processing; however, both a convection and a reaction term are also involved in the discussion. Under quite general assumptions, it is shown the presence of wavefront solutions and their main properties are studied. In particular, such wavefronts exist for every speed in a closed half-line and estimates of the threshold speed are given. The wavefront profiles are also strictly monotone and their slopes are uniformly bounded by the critical values of the diffusion. Joint work with Luisa Malaguti and Elisa Sovrano.

Università degli Studi di Ferrara
09/05/2022

Cristiana De Filippis
Quasiconvexity meets nonlinear potential theory

A classical problem in the regularity theory for vector-valued minimizers of multiple integrals consists in proving their smoothness outside a negligible set, cf. Evans (ARMA ’86), Acerbi & Fusco (ARMA ’87), Duzaar & Mingione (Ann. IHP-AN ’04), Schmidt (ARMA ’09). In this talk, I will show how to infer sharp partial regularity results for relaxed minimizers of degenerate, nonuniformly elliptic quasiconvex functionals, using tools from nonlinear potential theory. In particular, in the setting of functionals with (p,q)-growth - according to the terminology introduced by Marcellini (Ann. IHP-AN ’86; ARMA ‘89) - I will derive optimal local regularity criteria under minimal assumptions on the data. This talk is partly based on joint work with Bianca Stroffolini (University of Naples Federico II)

Università di Parma
16/05/2022

Eduard Feireisl
On Rayleigh-Benard problem in the framework of compressible fluid flows

We consider the physically relevant fully compressible setting of the Rayleigh-Benard problem of a fluid confined between two parallel plates, heated from the bottom, and subjected to the gravitational force. Under suitable restrictions imposed on the constitutive relations we show that this open system is dissipative in the sense of Levinson, meaning there exists a bounded absorbing set for any global-in-time weak solution. In addition, global-in-time trajectories are asymptotically compact in suitable topologies and the system possesses a global compact trajectory attractor. The standard technique of Krylov and  Bogolyubov then yields the existence of an invariant measure - a stationary statistical solution sitting on the global attractor. In addition, the Birkhoff--Khinchin ergodic theorem provides convergence of ergodic averages of solutions belonging to the attractor a.s. with respect to the invariant measure.

Charles University Prague
23/05/2022

Alessandro Carlotto

The study of free boundary minimal surfaces (namely: of critical points for the area functional in the category of relative cycles), which goes back at least to Courant, has played a distinguished role within the class of geometric variational problems for almost a century.
Yet, several fundamental questions remain open. For instance, is it possible to realise any orientable, compact surface with boundary as a free boundary minimal surface in the Euclidean unit ball? And, if so, are such realisations unique modulo ambient isometries? I will present significant advances on these two questions, with special focus on a (very recent) strong non-uniqueness result. In joint work with M. Schulz and D. Wiygul, we showed that the topology and symmetry group of a free boundary minimal surface in the Euclidean unit ball do not determine the surface uniquely: for any sufficiently large integer g there exist in the unit ball two distinct, properly embedded, free boundary minimal surfaces having genus g, three boundary components and symmetry group coinciding with the antiprismatic group of order 4(g+1). These constructions build on novel gluing techniques, relying on the analysis of suitable nonlinear elliptic problems.

ETH, Zürich

06/06/2022 François Hamel
TBA

Université
d'Aix-Marseille

27/06/2022 University of Copenhagen
04/07/2022

Jean Van Schaftingen
Ginzburg--Landau functionals on planar domains for a general compact vacuum manifold

Ginzburg-Landau type functionals provide a relaxation scheme to construct harmonic maps in the presence of topological obstructions. They arise in superconductivity models, in liquid crystal models (Landau-de Gennes functional) and in the generation of cross-fields in meshing. For a general compact manifold target space we describe the asymptotic number, type and location of singularities that arise in minimizers. We cover in particular the case where the fundamental group of the vacuum manifold in nonabelian and hence the singularities cannot be characterized univocally as elements of the fundamental group.
We obtain similar results for p-harmonic maps with p going to 2. The results unify the existing theory and cover new situations and problems.
This is a joint work with Antonin Monteil (Paris-Est Créteil, France), Rémy Rodiac (Paris–Saclay, France) and Benoît Van Vaerenbergh (UCLouvain).

UCLouvain

SEMINARI A.A. 2020/2021

 15/03/2021 Luca Rossi - https://meet.google.com/nie-attq-ged  Are solutions of reaction-diffusion equations asymptotically 1D ? The symmetry of solutions of elliptic equations is a classical and challenging problem in PDE, connected with stability. In this talk we are concerned with parabolic equations and we ask whether the 1-dimensional symmetry eventually emerges in the long time, for solutions which are initially non-symmetric. We will present a satisfactory answer in the case of the Fisher-KPP equation, together with some counter-examples and open questions. This topic is the object of a joint work with F. Hamel. Università di Roma Sapienza 29/03/2021 Long time behaviour of solutions to Hamilton-Jacobi equations for sub-Riemannian control systems Sub-Riemannian systems are an important class of nonlinear control systems with linear dependence on controls. Controllability properties for such systems are derived by the so-called Lie Algebra rank condition on the associated family of vector fields. We will discuss the long-time average behaviour of the value function of optimal control  problems for sub-Riemannian systems, which cannot be addressed by classical weak KAM theory as the Hamiltonian fails to be coercive in the momentum variable. Università di Roma Tor Vergata 26/04/2021 Free boundary minimal surfaces with connected boundary and arbitrary genus Besides their self-evident geometric significance, which can be traced back at least to Courant, free boundary minimal surfaces also naturally arise in partitioning problems for convex bodies, in capillarity problems for fluids and, as has significantly emerged in recent years thanks to work of Fraser and Schoen, in connection to extremal metrics for Steklov eigenvalues for manifolds with boundary (i. e. for eigenvalues of the corresponding Dirichlet-to-Neumann map). The theory has been developed in various interesting directions, yet many fundamental questions remain open. One of the most basic ones can be phrased as follows: does the Euclidean unit ball contain free boundary minimal surfaces of any given topological type? In spite of significant advances, the answer to such a question has proven to be very elusive. I will present some joint work with Giada Franz and Mario Schulz where we answer (in the affirmative) the well-known question whether there exist in B^3 (embedded) free boundary minimal surfaces of genus one and one boundary component. In fact, we prove a more general result: for any g there exists in B^3 an embedded free boundary minimal surface of genus g and connected boundary. This result provides a long-awaited analogue of the existence theorem obtained by Lawson in 1970 for closed minimal surfaces in round S^3. The proof builds on global variational methods, in particular on a suitable equivariant counterpart of the Almgren-Pitts min-max theory, and on a striking application of Simon's lifting lemma. ETH, Zürich 10/05/2021 Singularly-perturbed elliptic functionals: Γ-convergence and stochastic homogenisation In this talk we introduce a general class of singularly-perturbed elliptic functionals Fε and we study their asymptotic behaviour as the perturbation parameter ε > 0 vanishes. Under suitable assumptions, which in particular allow us to bound Fε by the Ambrosio-Tortorelli functionals, we show that the functionals Fε Γ-converge (up to subsequences) to a free-discontinuity functional of brittle type. Moreover, we provide asymptotic formulas for the limiting volume and surface integrands, which show that the volume and surface contributions of Fε decouple in the limit. If time permits, we will discuss the application of the general convergence result to the setting of stochastic homogenisation. This is joint work with R. Marziani and C. I. Zeppieri (Münster). Università di Roma Sapienza 07/06/2021 Existence and regularity of Faber-Krahn minimizers in a Riemannian manifold We consider the problem of finding domains that minimize the first eigenvalue of the Dirichlet Laplacian in a Riemannian manifold under volume constraint (Faber-Krahn minimizers). In the Euclidean setting such domains are balls, and existence and regularity of such domains is trivial. In a non-Euclidean setting very few examples are known. In this talk we will show a general result of existence and regularity of Faber-Krahn minimizers, inspired by the analogous result of existence and regularity of the solutions of the isoperimetric problems in a Riemannian manifold. In particular we will show that Faber-Krahn minimizers are regular in low dimension, and that there exists a critical dimension after which they can have singularities. Such critical dimension is related to the Alf-Caffarelli cone. This is a joint work with J. Lamboley. Universidad de Granada 21/06/2021 Henri Berestycki Segregation in predator-prey models and the emergence of territoriality I report here on a series of joint works with Alessandro Zilio (Université de Paris) about systems of competing predators interacting with a single prey. We focus on the analysis of stationary states, stability issues, and the asymptotic behavior when the competition parameter becomes unbounded. Existence of solutions is obtained by a bifurcation theory type approach and the segregation analysis rests on a priori estimates and a free boundary problem. We discuss the classification of solutions by using spectral properties of the limiting system. Our results shed light on the conditions under which predators segregate into packs, on whether there is an advantage to have such hostile packs, and on comparing the various territory configurations that arise in this context. These questions lead us to nonstandard optimization problems. EHESS, Paris

 07/10/2019 Università di Roma Tor Vergata 14/10/2019 Baruch College 21/10/2019 Universität Leipzig 28/10/2019 Universidad Tecnica Federico Santa Maria 04/11/2019 Università dell'Insubria 11/11/2019 Università di Torino 18/11/2019 Nicola Visciglia NLS: from the completely integrable to the general case Università di Pisa 25/11/2019 Francesco Clemente Effetto regolarizzante di un termine di ordine inferiore e proprietà di regolarità locale delle soluzioni in problemi di Dirichlet ellittici non coercitivi Sapienza Università di Roma 02/12/2019 Gabrielle Saller Nornberg On existence and multiplicity for a class of problems with natural gradient growth Sapienza Università di Roma 09/12/2019 Université Paris XIII 16/12/2019 Sapienza Università di Roma 20/01/2020 Université Paris Dauphine 27/01/2020 Universidad de Santiago de Chile 03/02/2020 Ning-An Lai  (ANNULLATO) Strauss exponent for semilinear wave equations with scattering space dependent damping Lishui University 10/02/2020 Universität Basel 17/02/2020 GSSI 24/02/2020 02/03/2020 09/03/2020 Jean Van Schaftingen   (ANNULLATO) Ginzburg-Landau functionals for a general compact vacuum manifold on planar domains Università di Ferrara  UCLouvain 16/03/2020 Xavier Ros-Oton  (ANNULLATO) TBA Universität Zürich 23/03/2020 Andrea Mondino  (ANNULLATO) TBA University of Oxford 30/03/2020 Giulio Ciraolo  (ANNULLATO) TBA Università di Milano Statale 06/04/2020 (ANNULLATO) TBA Institut National des Sciences  Appliquées de Rennes 20/04/2020 Martino Bardi     (ANNULLATO) TBA Università di Padova 29/04/2020 Lorenzo Dello Schiavo   (ANNULLATO) TBA Universität Bonn 04/05/2020 Sapienza Università di Roma 11/05//2020 Sapienza Università di Roma 18/05//2020 Seminario Sapienza Università di Roma 08/06//2020 Seminario Sapienza Università di Roma

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