Regular Lagrangian Flows, 20 years later
IIn this lecture I will survey the state of the art of the theory of Regular Lagrangian Flows, from by now classical results to the most recent developments, including classes of vector fields and ambient spaces where the theory is applicable.
Fire blocking problems: recent results and open questions
In this talk I will present the most recent result on the fire blocking problem: more precisely the problem is to construct a barrier with length proportional to time which the fire cannot cross, in order to block its spreading in finite time. The key parameter is the velocity of fire spreading and the velocity with which firefighters can construct the barrier.
In particular we will analyze the case where a single firefighter is building a barrier to block fire spreading: in this situation a necessary and sufficient condition can be given.
Kantorovich-Rubinstein duality theory for the Hessian and applications
The classical Kantorovich–Rubinstein duality theorem establishes a significant connection between optimal Monge transport and the maximisation of a linear form on 1-Lipschitz functions. This result is widely used in various fields of research, particularly in solid mechanics, where it is establishes a relationship between the geometric theory of Monge transport and the shape of certain structures in optimal design.
This talk presents a related theory in which the linear form is maximized on real functions of class $C^{1,1}$ whose spectral norm of the Hessian is less than 1, demonstrating that this maximisation is dual to a new three-marginal optimal transport problem in which the first two marginals are fixed and the third must dominate the other two in the sense of convex order. The existence of optimal transports enables solutions to the underlying Beckman problem to be expressed as a combination of rank-1 tensor measures supported by a graph. In the context of two-dimensional mechanics, this determines an optimal grillage which transfers a given load sytem. Another application is a new sharp functional inequality for distances in the space of probability measures on Euclidean space.
Topological singularities arising from non-local Ginzburg-Landau energies
We prove that suitably scaled functionals of Ginzburg-Landau type, given by the sum of quadratic fractional Sobolev seminorms and a penalization term vanishing on the unitary sphere, give rise to vortex-type singularities. A crucial step is the comparison with Ginzburg-Landau energies for Riesz potentials.
Work in collaboration with R. Alicandro, M. Solci, and G. Stefani
Understanding continuous solutions to balance laws
We consider continuous solutions u:(0,T)×R→R of scalar balance laws of the form u_t+f(u)_x=g with f:R→R given and smooth and g:(0,T)×R→R given and bounded. Depending on the nonlinearity of f, we mention what we mean by Lagrangian and Eulerian solutions, with surprises, and we prove better regularity results for u, Hölder and, along characteristics, Lipschitz. In the case of Burgers equation, when f(u)=u^2, we recover the well known Rademacher theorem for intrinsic Lipschitz graphs in the first Heisenberg group. Recent progress and future directions will be discussed. Based on joint works with Giovanni Alberti, Fabio Ancona, Stefano Bianchini, Francesco Bigolin, Francesco Serra Cassano, Alexander Cliffe, Elio Marconi, Andrea Pinamonti.
The Isoperimetric N-Cluster Problem: Classical Results and New Insights
I will present an overview of isoperimetric cluster problems, beginning with the classical isoperimetric problem and continuing to the most recent results of Neeman and Millman, which characterize minimizing clusters with three, four, and five chambers. Special attention will be given to the planar case and the hexagonal honeycomb theorem, which asserts that the hexagonal tessellation of the plane locally minimizes the perimeter among its compactly supported, unit-area variations. Finally, I will discuss a recent result obtained in collaboration with Francesco Maggi and Kenneth De Mason, providing a quantitative version of the planar honeycomb theorem.
Weak, renormalized, and vanishing-viscosity solutions of the two-dimensional Euler equations
Let us consider the Euler equations modeling the behavior of an incompressible, homogeneous, inviscid fluid. In the two-dimensional case, the Euler equations can be written in vorticity form as a continuity equation, in which the advecting velocity depends on the vorticity through an integral operator. In my talk, I will introduce several notions of weak solutions for the two-dimensional Euler equations in vorticity form: weak solutions, renormalized solutions, and vanishing-viscosity solutions. Relying on the linear theory for continuity equations with Sobolev velocity field by DiPerna and Lions, I will show that in the subcritical case weak solutions do not exhibit anomalies. In the supercritical case, I will show by means of a duality approach that the same holds for vanishing-viscosity solutions. This has some connections with the two-dimensional theory of turbulence of Kraichnan and Batchelor.
The Allard regularity problem
Stationary integral varifolds, introduced by Almgren in the sixties, are a very useful generalization of minimal surfaces, which play an important role in a variety of geometric problems. While all known examples of nonsmooth stationary integral varifolds consist of pieces of classical minimal surfaces coming together at a set of singularities of codimension 1, the only general regularity result available is the 1972 celebrated regularity theorem of Allard, which shows that the regular part of the varifold is dense in its support. Even proving that the singular part of 2-dimensional ones in R^3 has zero 2-dimensional measure is surprisingly challenging. In this talk I will explain what the difficulties are, propose some conjectures which we hope might simplify the problems, and present some partial results towards their solution, which anyway deliver some interesting structural consequences. The talk is based on two joint works with Camillo Brena, Stefano Decio, and Federico Franceschini.
Min-max construction of anisotropic minimal hypersurfaces
We use the min-max construction to find closed hypersurfaces which are stationary with respect to anisotropic elliptic integrands in any closed n-dimensional manifold . These surfaces are regular outside a closed set of zero n-3 dimension. The critical step is to obtain a uniform upper bound for density ratios in the anisotropic min-max construction. This confirms a conjecture posed by Allard. The talk is based on a joint work with A. De Rosa and Y. Li.
Locomotion and self-organization in biological and bio-inspired systems: recent results from the mechanics of active matter
Active matter is a broad field with many potential applications. A common thread underlying many of the current research lines is the study of systems powered by some internal energy source, as in the case of organisms moving thanks to food metabolism. In fact, self-propelling systems need to overcome the resistance of the surrounding medium, drawing the required energy from internal sources. The study of locomotion and self-organization in biological and bio-inspired artificial system appears, therefore, as an ideal testing ground to put the concepts and tools of active matter at work. We will report on recent progress coming from case-studies on the motility and collective feeding of unicellular organisms (flagellates and ciliates) and bio-inspired micro-robots, studied from the point of view of the mechanics of active matter.
From one-sided derivatives to full differentiability via stratification
We show that if a function admits one-sided derivatives then it is automatically differentiable out of a rectifiable set, and a corresponding stratification holds: the set where it is differentiable only along a k-subspace is k-rectifiable. This works for derivatives of any order and allows to deduce several known differentiability results, such as Rademacher's theorem, the rectifiablity of singular sets of convex functions, Alexandrov's theorem and the rectifiablity of the jump set.
The De Giorgi Conjecture for the Free Boundary Allen-Cahn Equation
The Allen-Cahn (AC) equation is known to approximate minimal surfaces, leading to the conjecture that global stable solutions to AC should be one-dimensional in dimensions up to 7. If true, this result would imply the celebrated De Giorgi conjecture for monotone solutions. Recognizing the interactive nature of the AC equation, Jerison advocated for two decades that a free-boundary version of AC would offer a more natural framework for approximating minimal surfaces. This perspective motivates studying the above conjecture within this free-boundary context. In recent joint work with Chan, Fernandez-Real, and Serra, we classify all stable global solutions to the one-phase Bernoulli free-boundary problem in three dimensions and, as a consequence, we establish that global stable 3d solutions to the free-boundary AC equation are one-dimensional.
Sharp interface for polycrystal: derivation and approximation
TBAI will present a phase-field approximation for sharp-interface variational models for grain boundaries in polycrystalline materials. The latter are energies defined on partitions that fully respect the symmetry of the underlying lattice. Individual grains are represented by the subsets of a partition of the domain and each subset has an orientation given by an element of the quotient space O(d)/G. The finite point group G ⊂ O(d) describes the symmetries of a reference lattice, and the local orientation is measured relative to this reference. The interface energy density has superlinear, $\theta|\log\theta |$, behaviour as the misorientation angle $\theta$ tends to zero. This behaviour represents the so called Read and Shockley law for small angle grain boundaries and it can be derived rigorously starting from a model of elasticity accounting for crystal defects. The phase-field approximation is a variant of the Ambrosio-Tortorelli scheme, proposed in a series of papers by Conti, Focardi and Iurlano in view of the application to cohesive fracture, and it is applicable to grain growth simulation and the reconstruction of grain boundaries from imaging data.
Uniqueness and functional convexity
Recently the notion of Functional convexity emerged, in different research projects for different reason. We'll discuss the meaning of this particular notion but also differences, and particular agreement with other convexity notions for vectorial problems
Stability for the surface diffusion flow
We study the surface diffusion flow in the flat torus, that is, smooth hypersurfaces moving with the outer normal velocity given by the Laplacian of their mean curvature. This model describes the evolution in time of interfaces between solid phases of a system, driven by the surface diffusion of atoms under the action of a chemical potential.
We show that if the initial set is sufficiently "close" to a strictly stable critical set for the Area functional under a volume constraint, then the flow actually exists for all times and asymptotically converges to a "translated" of the critical set. This generalizes the analogous result in dimension three, by Acerbi, Fusco, Julin and Morini.
Joint work with Antonia Diana e Nicola Fusco
Ambrosio-Tortorelli approach to topological singularities and connections with jump minimizing liftings
We study the Gamma-convergence of Ambrosio-Tortorelli-type functionals, for maps u defined on an open bounded set Ω ⊂ R^n and taking values in the unit circle S^1 ⊂ R^2. Depending on the domain of the functional, two different Gamma-limits are possible, one of which is nonlocal, and related to the notion of jump minimizing lifting, i.e., a lifting of a map u whose measure of the jump set is minimal. The latter requires ad hoc compactness results for sequences of liftings which, besides being interesting by themselves, also allow to deduce existence of a jump minimizing lifting. This is based on a joint work with Giovanni Bellettini and Riccardo Scala.
Layer potentials for elliptic operators with DMO-type coefficients: quantitative rectifiability and free boundary problems.
We investigate layer potentials for divergence-form elliptic operators L_A = -div(A∇) with coefficients A of Dini mean oscillation-type. First, we establish a quantitative rectifiability criterion: if a measure μ inside a ball is locally flat, has an L^2–bounded single-layer gradient with small oscillation locally and controlled density, then μ contains a large uniformly rectifiable piece inside that ball. Second, we prove a local big-pieces Tb theorem for such singular integrals under accretivity and weak boundedness conditions. Third, we apply these results to elliptic free-boundary problems: in Wiener-regular domains with Dini mean oscillation-type coefficients, obtaining quantitative and qualitative rectifiability for the one and two phase problems for operators with coefficients in the aforementioned regularity class. This is a joint work with M. Mourgoglou and C. Puliatti.
Nonuniformly elliptic regularity update
Nonuniformly elliptic equations have long been a central topic in the theory of partial differential equations, with foundational contributions dating back to the 1950s. The classical literature from the 1950s to the 1970s features seminal work by authors such as Finn, De Giorgi, Gilbarg, Stampacchia, Ladyzhenskaya, Uraltseva, Bombieri, Giusti, Miranda, Ivanov, Trudinger, Serrin, and Leon Simon. These equations are distinguished by a wide variation in the eigenvalues governing ellipticity which can lead to a failure of regularity for solutions (or minimizers, in the variational setting) unless appropriate structural conditions are imposed on this dispersion. More recent manifestations of nonuniform ellipticity arise in the modeling of strongly anisotropic materials [2], where the governing equations involve certain nonautonomous operators. In these contexts, standard techniques—such as perturbation methods— fail to deliver regularity results. Striking counterexamples emerge: even scalar, convex variational integrals with smooth integrands can admit minimizers that are no better than arbitrary competitors. In such cases, singularities may concentrate on fractal sets of maximal Hausdorff dimension. In my talk I will try to describe some recent advances in the regularity theory of such problems, including the setting of the longstading issue of absence of Schauder theory. The resulting techniques allow to treat new situations where standard types of ellipticity fail, like for instance those occurring in nonlinear homogenization problems, and lead to new approaches, changing the usual viewpoints. For instance, from a technical viewpoint, it is possible to derive the first direct approach to Lipschitz bounds without appealing to higher regularity, i.e., for the first time C0,1-estimates are derived without any appeal to C1,α-estimates in non-differentiable problems. The tools employed are different from those used in the past decades, and rely a delicate mix of nonlocal energy estimates in local problems, renormalization methods, use of nonlinear potentials of Havin-Mazya-Wolff type, insights from Harmonic Analysis like thermic charcterizations of Sobolev spaces and certain types of nonlinear Littlewood-Paley atomic type decompositions for Besov spaces.
From recent, joint work with Cristiana De Filippis (U Parma), a summary of which appears in [1].
References
[1] C. De Filippis, G. Mingione, Nonuniform ellipticity in variational problems and regularity, Notices of the Amer. Math. Soc., October 2025 issue.
[2] V. V. Jhikov, S. M. Kozlov, O. A. Olei¸nik, Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin xii+570 pp., (1994).
[3] L. Koch, M. Schäffner, Regularity for monotone operators and applications to homogenization of p-Laplace type equations. arXiv:2504.18401
Asymptotics for Ginzburg-Landau equations on complex line bundles over compact Riemannian manifolds
The Ginzburg-Landau equations represent a classical phenomenological model for superconductivity accounting for most observed effects, e.g. vorticity and magnetic flux quantization. They represent also a standard model in particle Physics, namely the abelian (or U(1)) Yang-Mills-Higgs model. Due to gauge invariance, this model has a natural formulation on complex line bundles. We carry out an asymptotic analysis when the Ginzburg-Landau parameter tends to infinity (i.e. in the so-called strongly repulsive limit) of minimizing configurations of the Ginzburg-Landau energy on (a given isomorphism class of) complex line bundles over a compact Riemannian manifold M, proving that energy densities concentrate on a codimension two integral cycle J which is mass minimizing in the homology class of M associated to that bundle, while the magnetic fields converge to a solution of the London equation with source term J. We prove also, exploiting a suitable monotonicity formula, that solutions to the Ginzburg-Landau equations within a logarithmic diverging energy regime converge to a codimension two stationary rectifiable varifold in M. This is joint work with Giacomo Canevari (Verona) and Federico Luigi Dipasquale (Scuola Superiore Meridionale, Naples).
On minimizing curves in a Brownian potential
We study a (1+1)-dimensional semi-discrete random variational problem that can be interpreted as the geometrically linearized version of the famous critical 2-dimensional random field Ising model (at zero temperature). We show that at every dyadic scale from the system size down to the lattice spacing the minimizer contains at most order-one Dirichlet energy per unit length, leading to a logarithmic scaling of the minimal energy. Using super-additivity in the scales, this allows us to establish a quenched homogenization result in the sense that he leading order of the minimal energy becomes deterministic as the ratio system size/lattice spacing diverges. This is joint work with Matteo Palmieri and Christian Wagner
Geometric Measure Theory for the evolution of dislocations
The evolution of dislocation loops under external and internal forces is a fundamental process at the core of the plastic deformation of crystalline solids. Because of the low general regularity that is to be expected of these loops, and the potential topological changes along the flow, integral and normal currents have long been identified the natural mathematical objects to model these phenomena. While the static theory of dislocation loops has been well understood for some time, the evolutionary theory is more recent. In this talk I will give an overview of some tools and results on how to model and analyze these geometric transport phenomena.
On the measure of the branch set of minimal immersions
I will present recent results on the structure of the singular set of a special class of stationary varifolds: namely, the branch set of an n-dimensional stationary varifold which can be locally represented as the graph of a C1,α two-valued function has locally finite measure and is rectifiable.
Characterizing pure unrectifiability via injectivity of projections
In this talk, we present a geometric characterization of Radon measures in Rn whose orthogonal projections onto (n-1)-planes are singular with respect to the (n-1)-dimensional Hausdorff measure, based on the so-called probabilistic injective projection property. Specifically, we prove that if a Radon measure μ is not supported on a single hyperplane, then it projects singularly with respect to H^(n-1) if and only if a typical orthogonal projection is injective on a set of full μ-measure. As a consequence, we obtain a characterization of codimension-one pure H^(n-1)-unrectifiability within the class of measures that disintegrate atomically with respect to orthogonal projections. In particular, if μ admits a non-trivial absolutely continuous component under projection along a positive measure set of directions, then μ must contain a non-trivial rectifiable part. This result can be viewed as a measure-theoretic analogue of the classical Besicovitch–Federer projection theorem, adapted to the setting of Radon measures. Finally, we show how this framework provides new insight into a longstanding conjecture of Marstrand concerning radial projections of purely unrectifiable sets with finite Hausdorff measure