Regularity of optimal sets for weighted and anisotropic isoperimetric problems.
Abstract: In this talk, we present some (old and new) regularity results for solutions to weighted and anisotropic isoperimetric problems. In particular, we deal with densities that are at most H\"older continuous (or merely continuous). As a consequence of the main result, we get the optimal regularity $C^{1, \frac{\alpha}{2-\alpha}}$, for certain anisotropic functionals.
The results are contained in joint works with. A Pratelli, L. Beck and C. Seis, B. Bulany and E.M. Merlino.
Soluzioni flat flow per il flusso per diffusione superficiale
Abstract: In un lavoro del 1994 J.W Cahn e J.E. Taylor hanno proposto uno schema variazionale per l’approssimazione delle soluzioni del flusso per diffusione superficiale mediante movimenti minimizzanti. Nel seminario mostreremo che tale schema permette effettivamente (almeno in due e tre dimensioni) di approssimare le soluzioni classiche dell’equazione per tutti i tempi in cui queste esistono.
The double spherical cap symmetrization and an application.
Abstract: This talk will be devoted to the discussion of an isoperimetric property of the double spherical cap symmetrization, that is, for any finite-perimeter set E in the plane, it holds $P(E^*)\leq P(E)+\mathcal{H}^1(\Gamma)$, where $E^*$ is the symmetrization of E and $\Gamma$ is a suitably defined set of bad radii. Additionally, I will show how the above property can be used to prove a property of the connected optimizers of the barycentric ratio, given by the isoperimetric deficit over the square of the barycentric asymmetry.
2D-lattice models for magnetic skyrmions under confinement
Abstract: Starting from 2D lattice spin configurations satisfying the uniform state e3 outside a bounded domain Ω, we investigate atomistic energies made of an exchange Heisenberg term together with an interfacial Dzyaloshinskii–Moriya (DMI) interaction.
A key ingredient of the analysis is the introduction of a discrete topological charge that avoids the degeneracy issues arising in the classical Berg–L ̈uscher construction and provides the right discrete analogue of the continuum degree. For sufficiently small DMI strength κ, we prove the existence of minimizers satisfying suitable energy barriers (lattice skyrmions) and the convergence, up to subsequences, to minimizers of the corresponding continuous energy (skirmions in the continuum). This result establishes a rigorous bridge between atomistic chiral spin systems and the continuum variational description of confined skyrmions in ultrathin ferromagnetic films.
Local boundedness of weak solutions to elliptic equations under unbalanced Orlicz growth conditions
Abstract: We present sufficient conditions for the local boundedness of local weak solutions to nonlinear elliptic equations of the form
− div A(x, ∇u) = f(x, u) in Ω,
featuring unbalanced growth. Specifically, the ellipticity and growth of the vector field A are governed by distinct Young functions — namely, non-negative convex functions vanishing at the origin — yielding an Orlicz-type setting. Our analysis relies on sharp Sobolev-type embeddings and a tailored construction of test functions, and is carried out without structural or symmetry assumptions on A. As a special case, this framework encompasses known results for equations with (p, q)-growth, refining them in certain limiting cases.
This talk is based on the preprint: “G. Giannone, Local boundedness of weak solutions to elliptic equations under unbalanced Orlicz growth conditions, 2025. arXiv:2510.18979”.
Curvature estimates for minimal hypersurfaces in the Heisenberg group
Abstract: In this talk we will examine minimal hypersurfaces in sub-Riemannian Heisenberg groups. We extend the celebrated Simons formula and Kato inequality to the sub-Riemannian setting, and we apply them to obtain integral curvature estimates for stable hypersurfaces. These results lead to structural conditions that imply a Bernstein-type rigidity theorem for smooth, non-characteristic hypersurfaces in the second Heisenberg group.
Enzo Maria Merlino
A Strong Form of the Quantitative Fractional Isoperimetric Inequality
Abstract:In this talk, we present a strong form of the quantitative fractional isoperimetric inequality. In particular, we show that the square root of the fractional isoperimetric deficit controls not only the Fraenkel asymmetry, but also a fractional-order oscillation of the boundary. This result can be viewed as the nonlocal counterpart of the local inequality established by Fusco and Julin. Our proof strategy relies on a regularization procedure for a volume-constrained shape-optimization problem driven by a nonlocal energy functional presenting both attractive and repulsive interactions. As an application, we will also discuss new stability estimates for a fractional Cheeger inequality. This is joint work with E. Cinti and B. Ruffini.
Existence and geometric properties of a constrained droplet in the half-space
Abstract: The study of equilibrium shapes of liquid droplets on a flat surface is a classical problem in capillary theory. In the standard model, droplets arise as minimizers of surface energy under a fixed volume constraint. In this talk, we consider a variant where the droplet is required to cover a prescribed region Ω on the supporting plane. We discuss the existence of minimizers, their geometric properties, and the behavior in the small-volume regime.
Aldo Pratelli
Proprietà di connessione dei minimal cluster di piccolo volume su varietà.
Abstract: In questo seminario discuteremo del problema di minimal cluster di piccolo volume su varietà. In particolare, ci chiederemo se tali cluster sono necessariamente connessi. Grazie a vari risultati recenti di diversi autori, vedremo che effettivamente è così nel caso di varietà Riemanniane, ma anche nel caso di varietà di tipo Finsler "a norma costante". Per varietà completamente generali questo risultato è falso, ma si può dare un bound al numero di zone connesse.
Isoperimetric sets and profile monotonicity in planar convex bodies
Abstract: Given a planar convex set K, we provide a complete characterization of the isoperimetric sets in K. The proof relies on a suitable Legendre-type duality with the prescribed-curvature functional, which also yields regularity properties of the isoperimetric profile. As an application, we prove that the isoperimetric profile is monotone along the volume-preserving parallel-body flow. This is based on joint work with Fattah, Ftouhi, and Leonardi.
On a multiphase vectorial Bernoulli free boundary problem.
Abstract: In this talk, we discuss the regularity of minimizers of a multiphase vectorial Bernoulli free boundary problem. This problem consists in a minimization problem for the Bernoulli functional over families of Sobolev functions with disjoint supports across distinct families. We prove that minimizers exist, are locally Lipschitz continuous, and that their free boundaries do not contain points where three or more phases meet. Our main regularity result establishes that the free boundary is locally a $C^{1,\eta}$ graph near two-phase and branching points for some $\eta \in (0,1)$. This is a joint work with Bozhidar Velichkov.
Fractional Poincarè Constants and Capacitary Notions of Inradius
Abstract: In this talk, we present estimates for the sharp constants in fractional Poincarè-Sobolev inequalities associated with an open set, in terms of a nonlocal capacitary extension of the inradius. Our approach is based on a new Maz’ya-Poincarè inequality and, as a further result, we also obtain new fractional Poincarè-Wirtinger-type estimates on balls. These inequalities display sharp limiting behaviors with respect to the fractional order of differentiability. Finally, we derive a new criterion for the embedding of the homogeneous Sobolev space Ds,p0 (Ω) into Lq(Ω), valid in the subcritical regime, as well as a characterization of the positivity of the fractional Cheeger constant. Based on joint project with Francesco Bozzola.