Given a cost function and two probability measures, the optimal transport problem is that of finding a transport map (or a plan) which minimises total cost. The case of finite-valued costs is well-understood and, under mild assumptions, the optimal plan has a special geometric structure. In particular, there exists a function, which we call a potential, whose c-subgradient contains the support of the optimal transport plan (for the quadratic cost \(|x-y|^2\) the gradient of the potential is famously known as the Brenier map). However, if a cost function attains infinite values, which corresponds to prohibiting certain pairs of points to be mapped to one another, only special families of costs were studied. We present a unified approach to transportation with respect to infinite-valued costs: we discuss compatibility of measures involved, give a sufficient condition for the existence of a Brenier-type map, and explain how this condition gives rise to abstract dualities on sets. The talk is based on joint work with S. Artstein-Avidan and S. Sadovsky.

We deal with planar systems involving an indefinite weight. We present a topological degree approach to prove the existence of periodic solutions which are positive in the first component. To achieve our goal, we need to introduce a suitable concept of superlinearity and also develop a specific form of the strong maximum principle. The talk is based on joint work with Juan Carlos Sampedro and Fabio Zanolin.

Nell'ambiente sub-Riemanniano del gruppo di Heisenberg, presenteremo uno studio sulla perdita di compattezza nell'immersione critica di Sobolev. In particolare, mediante tecniche variazionali, mostreremo una approssimazione energetica sottocritica in aperti limitati (non necessariamente regolari) e che l'energia delle relative funzioni ottimali si concentra in un unico punto. Assumendo naturali ipotesi di regolarità sul dominio, in linea col particolare ambiente geometrico sottostante, proveremo che il punto di concentrazione può essere localizzato mediante la funzione di Green, provando così che una congettura di Brezis e Peletier (Essays in honor of Ennio De Giorgi 1989) continua a valere anche per il gruppo di Heisenberg. La dimostrazione di tale risultato di localizzazione necessiterà di diversi risultati originali e indipendenti, come ad esempio un preciso controllo asintotico delle estremali tramite le funzioni ottimali di Jerison e Lee; stime di tipo Caccioppoli e Schauder per equazioni del tipo di CR Yamabe; la "Global Compactness" estesa sul gruppo di Heisenberg con una dimostrazione completamente nuova rispetto a quella nel lavoro originale di Struwe.

We consider a family of vectorial models for cohesive fracture, which may incorporate \(SO(n)\)-invariance. The deformation belongs to the space of (generalized) functions of bounded variation and the energy contains an (elastic) volume energy, an opening-dependent jump energy concentrated on the fractured surface, and a Cantor part representing diffuse damage. In recent work, together with Sergio Conti (U. Bonn) and Flaviana Iurlano (U. Sorbonne), we have shown that this type of functionals can be naturally obtained as \(\Gamma\)-limit of an appropriate phase-field model. The energy densities entering the limiting functional can be expressed, in a partially implicit way, in terms of those appearing in the phase-field approximation. Along the talk, we will comment on some phase-field models that have been introduced and analyzed since the seminal works of Ambrosio and Tortorelli and that finally have led to those we have proposed.

Abstract PDF.

There has been quite a lot of work done recently in various generalized models of random polytopes in convex bodies. One such model is when one takes n independent identically distributed uniform random points from a suitable convex body and considers the intersection of all congruent closed balls that contain the points. The resulting intersection is called a random ball-polytope. One is usually interested in the behaviour of both the metric properties of such objects, such as volume, surface area,intrinsic volumes, and also the combinatorial structure, like number of vertices, facets, etc. As in the case of classical random polytopes, this behaviour heavily depends on the smoothness of the boundary of the mother convex body. In this talk we discuss the behaviour of the vertex number and area of random disc-polygons, which is the two-dimensional case of the above model. Our aim is to prove series expansions for the expectation of the vertex number and area of random disc-polygons depending on the degree of smoothness of the boundary of the convex disc. Joint work with N. Montenegro (University of Szeged, Hungary).

We present an abstract method for studying the stability of parabolic flows, exploiting the Gamma-convergence of the corresponding energy functionals. We apply such a result to analyse the behavior of the s-fractional heat flows, as s tends to 0 and to 1, and of the s-Riesz flows, as s tends to 0 and to d (where d is the dimension of the ambient space). Time permitting, we present also the corresponding stability results for the geometric flows.

We consider a classical chemostat model of a one microbial species with a periodic input of a single nutrient having period \(\omega\), which is described by a system of differential delay equations. The delay represents the interval time between the consumption of the nutrient and its metabolization by the microbial species. In recent works, existence results of \(\omega\) - periodic solutions \((s(t),x(t))\) have been obtained, where \(s(t)\) and \(x(t)\) represent, respectively, the \(\omega\) - periodic evolution of the nutrient and of the microbial population. We present in this seminar a recent joint work with Pablo Amster (University of Buenos Aires) in which we show a global bifurcation behaviour of the solutions. The proof is topological and is based on the degree theory.

I discuss regularity results for solutions of nonuniformly elliptic equations. Moreover, I will discuss applications in stochastic homogenization and regularity for nonuniformly elliptic variational integrals.

Our main interest lies in the shape of a bounded domain for which a parametrized overdetermined boundary value problem admits a solution. Unlike a typical nonlinear problem where the non-degeneracy of the linearized operator implies a local one-to-one correspondence between parameters and solutions, overdetermined problems generally fail to follow this scenario because of a loss of derivatives. In this talk, I will explain a general perturbation result in overdetermined problems based on a characterization of an evolving domain by a geometric evolution equation. It turns out that the non-degeneracy and an additional monotonicity condition of the linearized operator are the properties inherited by the original problem, i.e., these linear properties imply the existence of a monotonically increasing family of domains admitting solvability of the corresponding overdetermined problem under a small continuous deformation of parameters.

In the talk I will consider the problem of finding the optimal shape of an obstacle which minimizes the drag force in an incompressible Stokes flow under Navier conditions at the boundary. I will propose a relaxation of the problem within the framework of free discontinuity problems, modeling the obstacle as a set of finite perimeter and the velocity field as a special function of bounded deformation (SBD): within this approach, the optimal obstacle may develop naturally geometric features of co-dimension 1.

In a celebrated three-pages long paper in 1979, Aronson and Bénilan obtained a remarkable estimate on second order derivatives for the solution of the porous media equation. Since its publication, the theory of porous medium flow has expanded relentlessly with applications including modeling of tissue growth and cancer development. In this talk, based on [1], we first motivate the use of the porous media equation to study the phenomenon of the tumor growth and then we present an extension of the Aronson-Bénilan estimate in \(L^\infty\) holding for any fields of pressure.

[1] G. Bevilacqua, B. Perthame, M. Schmidtchen; The Aronson-Bénilan estimate in Lebesgue spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire (2022), published online first, arxiv.org/pdf/2007.15267.pdf.

The Moser-Trudinger inequality is known as the critical case of the Sobolev inequality. In this talk, we propose a power type approximation of the Moser-Trudinger functional and discuss that its concentration level converges to the Carleson-Chang limit.

Fracture mechanics is nowadays a consolidated theory within the realm of continuum mechanics, that has been developed starting from the pioneering work of Griffith where the key concepts of energy release rate and fracture toughness (surface energy) has been established. A lacking feature of Griffith's fracture criterion, including its variational extension, is the inability to capture and describe fatigue cracks, that is cracks evolving due to repeated loads that singularly would be too small to cause the material failure. The aim of this contribution is to discuss how to possibly endow Griffith's fracture theory and related phase-field models with the ability to describe fatigue cracks. The proposed idea of the updated model is to consider the fracture toughness not as a material parameter anymore but rather as a material function, assumed to be decreasing as an accumulated energy based measure increases.

In this talk I shall recall a quite famous variational energy from quantum mechanics: the Gamow [liquid drop] model, and some questions about it (some solved, some open). Then I will consider two more energies: the first is the counterpart of the Gamow model with a spectral energy term instead of a surface energy one. The second is a reduced Hartree type energy. These energies have very similar behavior and quite similar mathematical formulations. For the latter two I will present some optimal design problems. Precisely, in some regimes one can show that the ball is a rigid minimizer. I will outline the strategy to do that for one of the two cases. The talk is based on the paper [MR] in collaboration with D. Mazzoleni (Pavia) and on a work in progress in collaboration with D. Mazzoleni and C. B. Muratov (Pisa).

[MR] Mazzoleni, Dario (I-PAVI); Ruffini, Berardo (I-BOLO): A spectral shape optimization problem with a nonlocal competing term. (English summary) Calc. Var. Partial Differential Equations 60 (2021), no. 3, Paper No. 114, 46 pp.

In this talk I will introduce a model for linear elasticity on deformations showing cracks. The elastic energy plus a penalization energy of the fractured zone due to the atomic interaction is considered and a Gamma convergence result for this is enstablished, showing the arising of topological singularities in an otherwise perfect lattice. The approach is reminescent of the one for Ginzburg Landau and screw dislocations models, but avoiding the core radius technique. To deal with this a new general notion of distributional determinant for BV maps is introduced.

Several differential equations arising from models of real phenomena can be rewritten as semilinear differential equations in function spaces. In this talk, we show some existence results for semilinear differential or integro–differential inclusions in Banach spaces, which turn out to be a unified method for studying some physical, biological and engineering models. Our apploach is based on multivalued analysis and fixed point theory.

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.

In the effort of variationally solving problems in fracture mechanics, damage gradient models have proven to be very useful in the numerical approximation of the Griffith functional [Ambrosio, Tortorelli: 1990]. Within these models the (scalar) damage field \(\alpha\in [0,1]\) gives a coarse description of the progressive material damaging; it modulates, as a whole, the stiffness tensor and its evolution is associated with a suitable dissipation.
I will discuss some generalization of these phase-field models where the selective modulation of the components of the stiffness tensors allows for the effective description of new classes of phenomena. In particular I shall focus on some problems of material remodelling where the internal variable is actually an angle describing the orientation of emerging preferential directions. I will also discuss the case of structural theories of beams and shells where the modulation of the elastic energy as a whole, used by several authors, has no physical evidence. From the mathematical point of view, small variations from the standard Ambrosio Tortorelli assumptions, such as the non convexity of the energy with respect to the internal variables, open new questions when analyzing the Gamma–convergence of these models.

The talk deals with the regularity problem for minimizers of an integral functional with nonstandard growth conditions. We present Lipschitz continuous results for the local minimizers of the energy function with general slow growth and some examples of applications are exhibited. Moreover, we give some results of existence and higher differentiability for a class of problems under \(p, q\) sub–quadratic growth. Finally, we provide a first approach to the study of regularity for degenerate problems.
The researches are developed in collaboration with Michela Eleuteri, Paolo Marcellini, Stefania Perrotta, Giovanni Cupini e Antonella Passarelli di Napoli.

We discuss a natural generalization of the classical Blachke-Santaló inequality for \(n>2\) sets and functions, related multimarginal Kantorovich problem, entropy estimates and Monge-Ampère type equations. This problem was motivated by the geodesic barycenter problem from the optimal transportation theory.

We review recent results on critical points of the sub-Finsler area in the first Heisenberg group, including Pansu-Wulff shapes, regularity of surfaces with prescribed mean curvature, existence of Lipschitz solution for CMC graphs, minimizing cones and Bernstein type problems. These results have been obtained in collaboration with A. Pinamonti, J. Pozuelo, M. Ritoré, and S. Verzellesi.

In this talk we present a notion of weak solution for the wave equation on a time-dependent domain with homogeneous Dirichlet boundary value data and standard initial conditions. Under regularity assumptions of the time evolution of the space domain, existence can be proven via penalization argument (cf. [4]) or via time discretization and approximation argument by cylindrical domains (cf. [1]).
Moreover, in the case of a \(C^{1,1}\)-evolution, by means of suitable diffeomorphisms, the problem can be recast into an hyperbolic equation on a fixed domain: we prove the equivalence of the notions of weak solutions of the two problems.
Finally, this equivalence allows us to obtain an energy balance, following a double regularization argument in the spirit of [2], and to discuss, by means of a standard Galerkin method, better regularity properties depending on the evolution of the domain and the initial data.
As an application, we give a rigorous definition of dynamic energy release rate density for some problems of debonding, and we formulate a proper notion of solutions for such problems.
This talk is based on a joint work with G. Lazzaroni, F. Riva, and F. Solombrino (cf.[3]).

Keywords: hyperbolic equation in time-dependent domains, energy balance, debonding model.

References:
[1] J. Calvo, M. Novaga and G. Orlandi: Parabolic equations in time-dependent domains, J. Evol. Equ., 17 (2017), pp.781-804.
[2] G. Dal Maso, and I. Lucardesi: The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data, Appl. Math. Res. Express. AMRX, 1 (2017), pp. 184-241.
[3] G. Lazzaroni, R. Molinarolo, F. Riva, and F. Solombrino: On the wave equation on moving domains: regularity, energy balance and application to dynamic debonding, Accepted.
[4] J.P. Zolésio: Galerkin approximation for wave equation in moving domains, Stabilization of Flexible Structures, Lecture Notes in Control and Information Sciences, 147, Springer (1990).

In this talk we discuss the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.
Joint work with F. Cagnetti, G. Dal Maso, and L. Scardia.