Dipartimento di Matematica
Guido Castelnuovo

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. 

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).

 Universidade de São Paolo 

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.

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.

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).

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.                                                                                                                                                                                                                   

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.

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

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.

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).

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.

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. 

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. 

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

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.

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. 

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)

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.

Alessandro Carlotto
A strong topological non-uniqueness result for free boundary minimal surfaces

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.

François Hamel
Symmetry properties for the Euler equations and related semilinear elliptic equations

In this talk, I will discuss radial and one-dimensional symmetry properties for the stationary incompressible Euler equations in dimension 2 and some related semilinear elliptic equations. I will show that a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions in an annulus is necessarily a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured plane, with some suitable conditions at infinity or at the origin. If possible, I will also discuss the case of parallel flows in two-dimensional strips, in the half-plane and in the whole plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on radial and one-dimensional symmetry results for the solutions of some elliptic equations satisfied by the stream function. The talk is based on joint works with N. Nadirashvili.

Niels Martin Moeller
Universal entropy bounds for embedded self-shrinkers with symmetries

We consider complete self-shrinking solitons for the mean curvature flow in R^(n+1), meaning minimal hypersurfaces with respect to a Gaussian conformal background metric. Using comparison geometry, we prove that there is a universal constant bounding the entropies of all such embedded shrinkers with a rotational symmetry. As an application, we prove smooth compactness within this class. Finally, we also show that if we impose an additional reflection symmetry, then in each dimension there are only finitely many such shrinkers. This is joint work with Ali Muhammad and John Ma.

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).