Giovanni Alberti
Title: Construction of pathological sets via Baire theorem
Abstract: In this talk I will consider a few (more or less) known classes of "pathological" sets that play a role in GMT and beyond, and named after Besicovitch, Davies and Larman, who produced the first examples of such sets. I will present a construction based on Baire theorem (following T.W. Körner) which is unifying to a certain extent, and gives examples that, to my knowledge, are new. Finally I will discuss some open problems of similar flavour. This is joint work with Jeremy Mirmina (University of Pisa).
Martina Bellettini
Title: A model of anisotropic branched optimal transport
Abstract: We introduce a possible model for a branched optimal transport problem requiring that the transport from a source to a target happens in an anisotropic space. More precisely, we define a general anisotropic transport cost on directed weighted graphs and we characterize its relaxation on the space of normal 1-currents. This allows one to prove the existence of minimizers. The characterization relies on an integralgeometric equality for anisotropic integrands, which will be the main focus of the talk.
Giulia Bevilacqua
Title: Classical solutions for a soap film capillarity problem
Abstract: In this talk, we study the regularity of the soap film capillarity problem, that is soap films which are chosen to be sets of finite perimeter containing an assigned amount of volume and satisfying a topological spanning condition. For plane boundaries, we show that these minimizers are normal smooth graphs with positive constant mean curvature and meeting orthogonally the boundary. Joint work with Salvatore Stuvard (UNIMI) and Bozhidar Velichkov (UNIPI).
Gianmarco Caldini
Title: On the smooth approximation of integral cycles
Abstract: In this seminar I will explain how closely one can approximate an integral current representing a given homology class by means of a smooth submanifold. This is a joint study with William Browder and Camillo De Lellis, based on some previous preliminary work of the former author together with Frederick Almgren.
Vito Crismale
Title: Adaptive finite element approximation for quasistatic crack growth
Abstract: We approximate quasistatic evolutions for Brittle Fractures in 2d by evolutions that are discrete both in time and in space. More precisely for any time the discretized energies are defined on functions piecewise affine on an adaptive triangulation with given mesh size, vanishing with the time step. The result is based on a void modification lemma applied to the union of 'jump triangles' where the symmetrized gradient is large. The work is in collaboration with M. Friedrich and J. Seutter (FAU Erlangen).
Luigi De Masi
Title: Regularity of capillary minimal surfaces
Abstract: In this talk I present an Allard-type regularity theorem for minimal surfaces meeting the boundary of a container with a prescribed angle, obtained in collaboration with N. Edelen, C. Gasparetto and C. Li: if a stationary surface for a capillarity functional is close to a half-plane in a measure-theoretic sense, then it is a $C^{1,\alpha}$-graph over the half-plane with uniform estimates.
The proof relies on a "boundary length" control (a fact with its own interest) and viscosity techniques.
Giacomo Del Nin
Title: From one-sided derivatives to full differentiability: a stratification approach
Abstract: We present a general stratification result that shows that, whenever a function $u$ on $\R^n$ admits one-sided derivatives at every point, then it is fully differentiable out of an $(n-1)$-rectifiable set, and a corresponding stratification holds: the function is differentiable along a $k$-dimensional subspace out of a $(k-1)$-rectifiable set.
This also works for higher-order differentiability: if $u$ admits at every point a “homogeneous” Taylor expansion up to order $m$ (that is, the last term is just $m$-homogeneous instead of being a polynomial) then the homogeneity is automatically upgraded to polynomiality outside of an $(n-1)$-rectifiable set, and a similar stratification holds.
We then show how this principle can be applied to derive several seemingly unrelated differentiability results, such as: the $k$-rectifiability of the $k$-th singular stratum of convex functions and of distance functions; the second-order differentiability of convex functions; and the rectifiability of the jump set.
Marco Di Marco
Title: Stokes’ Theorem in Heisenberg groups
Abstract: We introduce the notion of submanifolds with boundary with intrinsic C^1 regularity in the setting of sub-Riemannian Heisenberg groups. We present a Stokes’ Theorem for such submanifolds involving the integration of Heisenberg differential forms introduced by Rumin. The talk is based on a joint work with A. Julia, S. Nicolussi Golo and
D. Vittone.
Alberto Fiorini
Title: Minimality of lamellae for the anisotropic Ohta-Kawasaki energy: a second variation approach
Abstract: We consider a different version of the Ohta-Kawasaki energy obtained by replacing the perimeter term with the anisotropic perimeter, and study its minimization on the bidimensional torus. In particular we focus on proving that horizontal strips are local minimizers (w.r.t. some suitably defined C^2 distance) by following an approach devised by E. Acerbi, N. Fusco, M. Morini in the isotropic case.
Carlo Gasparetto
Title: Boundary regularity for singular minimal surfaces
Abstract: In this talk, I will address some regularity properties of almost minimizers of a perimeter functional with a weight that degenerates at the boundary of a domain. These objects have connections with free boundary problems and with classical minimal surfaces with rotational symmetries.
Based on a joint work with Filippo Paiano and Bozhidar Velichkov.
Virginia Lorenzini
Title: On the regularity of minimizers of an Ambrosio-Tortorelli functional with linear growth and of its $\Gamma$-limit
Abstract: In the one-dimensional setting we consider an Ambrosio-Tortorelli type functional of $F_\eps(u,v)$ which has linear growth with respect to $u'$. We prove that under suitable conditions on the fidelity term, $F_\eps$ admits Sobolev regular minimizers, and that the same is true for the $\Gamma$-limit $F$ of $F_\eps$. As a byproduct, we show that the functional $A(u)$ computing the length of the generalized graph of a function of bounded variation $u$, under the same conditions on the fidelity term, admits a unique minimizer that has to be Sobolev regular. This solves a Conjecture of De Giorgi in the one dimensional case.
Valentino Magnani
Title: On the spherical measure of regular subsets in homogeneous groups.
Abstract: We present an overview about the problem of computing the spherical measure of different classes of regular sets in homogeneous groups. We finally highlight some recent results and developments.
Elio Marconi
Title: Hölder regularity for balance laws and intrinsic Lipschitz graphs
Abstract: We consider continuous solutions $u:(0,T)\times \mathbb{R} \to \mathbb{R}$ of scalar balance laws of the form $u_t+f(u)_x=g$ with $f:\mathbb R \to \mathbb R$ given and smooth and $g \in L^\infty((0,T)\times \mathbb R)$. Depending on the nonlinearity of $f$ we prove Hölder regularity results for $u$: in the case of Burgers equation ($f(u)=u^2$), we recover the known Rademacher theorem for intrinsic Lipschitz graphs in the Heisenberg group. Recent progresses on more general Carnot groups will be discussed. This is a joint work with Laura Caravenna and Andrea Pinamonti.
Annalisa Massaccesi
Title: Constructions for a C^1 function with prescribed gradient on a Cantor-type set
Abstract: In this talk I will outline the iterative construction of a C^1 function u, with \|u\|_\infty \leq \eta and Du(x)=F(x,u(x)) on a Cantor-type set C. It is transparent from the construction the presence of a trade off between the size of C and the Hölder regularity of Du. This type of construction is the building block for counterexamples to Frobenius theorem when the tangency set is not regular enough.
Andrea Merlo
Title: On Sets with Unit Density in Homogeneous Groups
Abstract: In this talk, I will present a result obtained in collaboration with A. Julia, which establishes that in every homogeneous group, there exists a metric such that any Borel set with unit density (with respect to the Hausdorff measure defined by this metric) is Euclidean rectifiable. In other words, such sets can be covered up to a null set by a countable union of Lipschitz images of compact subsets of R^n. The converse also holds thanks to a result by Kirchheim. This result extends to homogeneous groups a celebrated theorem of Mattila from 1975.
Emanuele Paolini
Title: Infinite Isoperimetric Partitions
Abstract: I would like to present the results obtained in three recent papers obtained in collaboration with Novaga-Stepanov-Tortorelli, Novaga and Novaga-Nobili about three different kinds of infinite isoperimetric partitions. An isoperimetric partition is a way to subdivide a space in many different regions with prescribed measure so that the common interface between the regions has the smallest possible measure. The three proposed examples are "infinite" in three somewhat different ways.
In the first case we consider infinitely many regions which become smaller and smaller so that the total measure is finite. In this case we have some examples and a very weak regularity result in the planar case.
In the second case we consider a finite (small) number of regions but some of the regions might have infinite measure: we are able to reduce this problem to the usual isoperimetric problem for bubble clusters and hence obtain a description of the isoperimetric partitions in the cases when the corresponding problem for bubble clusters has been solved.
In the third case (which is a work in progress) we consider infinite periodic tilings of the space with a finite number of tiles with prescribed measure: we are able to find the solution in the case of two tiles (with any given prescribed areas) in the plane.
Riccardo Scala
Title: On the minimal jump set lifting problem and its Ambrosio-Tortorelli approximation
Abstract: We consider the Ambrosio-Tortorelli functional (AT) for maps $u_k$ with values in the unit circle, in two settings with different hypotheses. When the involved maps $u_k$ are constrained to stay in the $H^1$ space, the AT functionals $Gamma$-converge to a functional counting the minimal length of the jump set of a lifting of the limiting map $u$. We show that it is possible to find such a minimal lifting and how it is related to a branched transportation problem. This is a joint work with Roberta Marziani and Giovanni Bellettini.
Giorgio Stefani
Title: Sharp conditions for the BBM formula and application to heat content asymptotics
Abstract: We give sufficient and necessary conditions for the validity of the Bourgain-Brezis-Mironescu (BBM) formula. These results are applied to determine the asymptotics of the relative heat content in both local and nonlocal settings. This is a joint work with L. Gennaioli.
Anna Skorobogatova
Title: The regularity of area-minimizing currents modulo p: part II
Abstract: A natural framework in which to study the Plateau problem is by using currents with multiplicities modulo p, for a fixed integer p. This setting allows for area-minimizing surfaces to exhibit codimension 1 singularities like triple junctions, and has close connections to the known regularity theory for stable minimal surfaces.
In this talk, which is a continuation of the talk by Salvatore Stuvard on the regularity of codimension 1 mod(p) area-minimizing currents, I will focus on the regularity for higher codimension mod(p) area-minimizers. I will discuss joint work with Camillo De Lellis and Paul Minter where we establish a structural result on the singular set, which is an analogue of that known for hypersurfaces. I will emphasize the difficulties that arise here in contrast to the codimension 1 setting, and some interesting remaining open questions.
Salvatore Stuvard
Title: The regularity of area-minimizing currents modulo p: part I
Abstract: Area-minimizing currents modulo p, where p≥2 is an integer, are integer rectifiable currents solving Plateau’s problem in their Z_p homology class. Compared to area-minimizers in the integral sense, they exhibit a far richer geometric complexity, often observed in soap films: for example, interior singularities may be present in low dimension even if the codimension is 1. If p is an even integer, such singularities may be branched.
The aim of this talk is to present and discuss some results providing a rather complete partial regularity theory for minimizers modulo p: this includes dimensional estimates for singular sets in arbitrary codimension as well as structure theorems in codimension 1. Structure theorems in higher codimension will be discussed in part II by Anna Skorobogatova.
Based on joint works with De Lellis (IAS), Hirsch (U Leipzig), Marchese (U Trento), and Spolaor (UC San Diego).
Emanuele Tasso
Title: A general criterion for slicing the jump set of a function
Abstract: In this talk a novel criterion for the slicing of the jump set of functions is presented, which overcomes the limitation of the classical approach based on a combination of codimension-one slices and the parallelogram law. This latter technique was originally developed for functions of bounded deformation and more recently adapted to the case of functions of bounded A-variation . Our method instead, builds upon a recent rectifiability result of integral geometric measures and extends to Riemannian manifolds, where slicing is conducted along geodesics. As a particular application, we derive the structure of the jump set of functions with (generalized) bounded deformation in a Riemannian setting. Furthermore, if time permits, I will show the critical role these structural results play in deriving certain non-local to local free-discontinuity problems.
Riccardo Tione
Title: On the Lawson-Osserman conjecture
Abstract: In 1977, H.B. Lawson and R. Osserman conjectured that Lipschitz maps which are critical with respect to outer variations of the area functional are also critical with respect to domain variations. In this talk I will present a solution to this conjecture in the planar case. This result was obtained in collaboration with J. Hirsch and C. Mooney.
Simone Verzellesi
Title: Rigidity results for complete, stable hypersurfaces in sub-Riemannian Heisenberg groups
Abstract: The Bernstein problem consists in characterizing global minimizers of suitable perimeter functionals. Roughly speaking, is it true that boundaries of global perimeter minimizers are flat? Solving this problem in the Euclidean setting has been a major driving force in the development of geometric measure theory and geometric analysis. While the Euclidean issue is well understood, the Bernstein problem in sub-Riemannian Heisenberg groups leaves many interesting questions still unanswered. In this talk, after a survey of the known results, we propose a solution in the second Heisenberg group. To this end, we introduce some relevant tools in the study of minimal surfaces, such as the sub-Riemannian analogues of Simons’ and Kato’s inequalities, and we discuss how to combine them to approach our problem. This talk is based on joint works with Gianmarco Giovannardi and Andrea Pinamonti.
Kilian Zambanini
Title: Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
Abstract: Stepanov differentiability theorem is a generalization of the classical Rademacher’s Theorem for Lipschitz functions between Euclidean spaces. We extend such a result in the subriemannian setting of Heisenberg groups, using the intrinsic notions of Lipschitz continuity and differentiability introduced by Franchi, Serapioni, and Serra Cassano. This result is based on the corresponding Rademacher’s theorem recently proved by Vittone. Joint work with M. Di Marco, D. Vittone and A. Pinamonti.