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Marco Caroccia

Publications and source records attributed to Marco Caroccia.

18 recordsLinked to original sources
arXiv2025

Rigidity and functional properties of $\mathrm{BD}_{dev}(Ω)$

We provide a structural analysis of the space of functions of bounded deviatoric deformation, $\mathrm{BD}_{dev}$, which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for $\mathrm{BD}_{dev}$-maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation and homogenization problems, allowing for integrands with explicit dependence on $u$ as well as $\mathcal{E}_d u$. Our approach overcomes several difficulties as compared to the $\mathrm{BD}$ case, in particular due to the lack of invariance of $\mathcal{E}_d$ under orthogonalization of the polar directions. Applications to integral representation and Material science are discussed.

math.AP↗
arXiv2025

Iterative blow-ups for maps with bounded $\mathcal{A}$-variation: a refinement, with application to $\mathrm{BD}$ and $\mathrm{BV}$

We refine the iterated blow-up techniques. This technique, combined with a rigidity result and a specific choice of the kernel projection in the Poincaré inequality, might be employed to completely linearize blow-ups along at least one sequence. We show how to implement such argument by applying it to derive affine blow-up limits for $\mathrm{BD}$ and $\mathrm{BV}$ functions around Cantor points. In doing so we identify a specific subset of points - called totally singular points having blow-ups with completely singular gradient measure $D p=D^s p$, $\mathcal{E} p=\mathcal{E}^s p$ - at which such linearization fails.

math.AP↗
arXiv2025

On the emergence of almost-honeycomb structures in low-energy planar clusters

Several commonly observed physical and biological systems are arranged in shapes that closely resemble an honeycomb cluster, that is, a tessellation of the plane by regular hexagons. Although these shapes are not always the direct product of energy minimization, they can still be understood, at least phenomenologically, as low-energy configurations. In this paper, explicit quantitative estimates on the geometry of such low-energy configurations are provided, showing in particular that the vast majority of the chambers must be generalized polygons with six edges, and be closely resembling regular hexagons. Part of our arguments is a detailed revision of the estimates behind the global isoperimetric principle for honeycomb clusters due to Hales (T. C. Hales. The honeycomb conjecture. Discrete Comput. Geom., 25(1):1-22, 2001).

math.OC↗
arXiv2024

On the singular planar Plateau problem

Given any $Γ=γ(\mathbb{S}^1)\subset\mathbb{R}^2$, image of a Lipschitz curve $γ:\mathbb{S}^1\rightarrow \mathbb{R}^2$, not necessarily injective, we provide an explicit formula for computing the value of \[ \mathcal A(γ):=\inf\left\{\left. \int_{B_1(0)}|\mathrm{det}(\nabla u)| \mathrm{d} x \ \right| \ u=γ\text{ on }\mathbb{S}^1\right\}, \] where the infimum is evaluated among all Lipschitz maps $u:B_1(0)\rightarrow \mathbb{R}^2$ having boundary datum $γ$. This coincides with the area of a minimal disk spanning $Γ$, i.e., a solution of the Plateau problem of disk type for the oriented contour $Γ$. The novelty of the results relies in the fact that we do not assume the curve $γ$ to be injective and our formula allows for any kind of self-intersections

math.AP↗
arXiv2023

Isoperimetric sets and $p$-Cheeger sets are in bijection

Given an open, bounded, planar set $Ω$, we consider its $p$-Cheeger sets and its isoperimetric sets. We study the set-valued map $\mathfrak{V}:[\frac12,+\infty)\rightarrow\mathcal{P}((0,|Ω|])$ associating to each $p$ the set of volumes of $p$-Cheeger sets. We show that whenever $Ω$ satisfies some geometric structural assumptions (convex sets are encompassed), the map is injective, and continuous in terms of $Γ$-convergence. Moreover, when restricted to $(\frac 12, 1)$ such a map is univalued and is in bijection with its image. As a consequence of our analysis we derive some fine boundary regularity result.

math.AP↗
arXiv2022

A Compactness Theorem for functions on Poisson point clouds

In this work we show a compactness Theorem for discrete functions on Poisson point clouds. We consider sequences with equibounded non-local $p$-Dirichlet energy: the novelty consists in the intermediate-interaction regime at which the non-local energy is computed.

math.AP↗
arXiv2022

Asymptotic behavior of the Dirichlet energy on Poisson point clouds

We prove that quadratic pair interactions for functions defined on planar Poisson clouds and taking into account pairs of sites of distance up to a certain (large-enough) threshold can be almost surely approximated by the multiple of the Dirichlet energy by a deterministic constant. This is achieved by scaling the Poisson cloud and the corresponding energies and computing a compact discrete-to-continuum limit. In order to avoid the effect of exceptional regions of the Poisson cloud, with an accumulation of sites or with "disconnected sites", a suitable "coarse-grained" notion of convergence of functions defined on scaled Poisson clouds must be given.

math.AP↗
arXiv2021

Dimensional lower bounds for contact surfaces of Cheeger sets

We carry on an analysis of the size of the contact surface of a Cheeger set $E$ with the boundary of its ambient space $Ω$. We show that this size is strongly related to the regularity of $\partial Ω$ by providing bounds on the Hausdorff dimension of $\partial E\cap \partialΩ$. In particular we show that, if $\partial Ω$ has $C^{1,α}$ regularity then $\mathcal{H}^{d-2+α}(\partial E\cap \partialΩ)>0$. This shows that a sufficient condition to ensure that $\mathcal{H}^{d-1}(\partial E\cap \partial Ω)>0$ is that $\partial Ω$ has $C^{1,1}$ regularity. Since the Hausdorff bounds can be inferred in dependence of the regularity of $\partial E$ as well, we obtain that $Ω$ convex, which yields $\partial E\in C^{1,1}$, is also a sufficient condition. Finally, we construct examples showing that such bounds are optimal in dimension $d=2$.

math.AP↗
arXiv2020

On the integral representation of variational functionals on $BD$

Following the global method for relaxation we prove an integral representation result for a large class of variational functionals naturally defined on the space of functions with Bounded Deformation. Mild additional continuity assumptions are required on the functionals.

math.AP↗
arXiv2020

On the Gamma convergence of functionals defined over pairs of measures and energy-measures

A novel general framework for the study of $Γ$-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the $Γ$-limit of these kind of functionals by knowing the $Γ$-limit of the underlining energies. In particular, the interaction between the functionals and the underlining energies results, in the case these latter converge to a non continuous energy, in an additional effect in the relaxation process. This study was motivated by a question in the context of epitaxial growth evolution with adatoms. Interesting cases of application of the general theory are also presented.

math.AP↗
arXiv2019

Mumford-Shah functionals on graphs and their asymptotics

We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford-Shah functionals converge to a minimizer of a continuum Mumford-Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate minimizers of the graph Mumford-Shah functional.

math.AP↗
arXiv2018

Equilibria configurations for epitaxial crystal growth with adatoms

The behavior of a surface energy $\mathcal F(E,u)$, where $E$ is a set of finite perimeter and $u\in L^1(\partial^* E, \mathbb R_+)$ is studied. These energies have been recently considered in the context of materials science to derive a new model in crystal growth that takes into account the effect of atoms freely diffusing on the surface (called adatoms), which are responsible for morphological evolution through an attachment and detachment process. Regular critical points, existence and uniqueness of minimizers are discussed and the relaxation of $\mathcal F$ in a general setting under the $L^1$ convergence of sets and the vague convergence of measures is characterized. This is part of an ongoing project aimed at an analytical study of diffuse interface approximations of the associated evolution equations.

math.AP↗
arXiv2017

The Cheeger N-problem in terms of BV functions

We reformulate the Cheeger N partition problem as a minimization among a suitable class of BV functions. This allows us to obtain a new existence proof for the Cheeger-N-problem. Moreover, we derive some connections between the Cheeger-2- problem and the second eigenvalue of the 1-Laplace operator.

math.AP↗
arXiv2016

On the isoperimetric properties of Planar N-clusters

This Thesis aims to highlight some isoperimetric questions involving the, so-called, $N$-clusters. We first briefly recall the theoretical framework we are adopting. This is done in Chapter one. In chapter two we focus on the standard isoperimetric problem for planar $N$-cluster for large values of $N$ and we provide an equidistribution energy-type results under some suitable assumption. The third Chapter is devoted to a stability results of the hexagonal honeycomb tiling. Finally in the fourth Chapter we consider a generalization of the Cheeger constant, defined as a minimization of a suitable energy among the class of the $N$-clusters. We show how this problem is related to the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian introduced by Caffarelli and Fang-Hua Lin in 2007. We conclude, in Chapter five, with some remarks and some possible future direction of investigation.

math.AP↗
arXiv2015

Cheeger $N$-clusters

In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the $N$-clusters contained in an open bounded set $Ω$. Here with $N$-Cluster we mean a family of $N$ sets of finite perimeter, disjoint up to a set of null Lebesgue measure. We call any $N$-cluster attaining such a minimum a Cheeger $N$-cluster. Our purpose is to provide a non trivial lower bound on the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian. Here we discuss the regularity of Cheeger $N$-clusters in a general ambient space dimension and we give a precise description of their structure in the the planar case. The last part is devoted to the relation between the functional introduced here (namely the $N$-Cheeger constant), the partition problem for the first Dirichlet eigenvalue of the Laplacian and the Caffarelli and Lin's conjecture.

math.AP↗
arXiv2014

A note on the stability of the Cheeger constant of $N$-gons

The regular $N$-gon provides the minimal Cheeger constant in the class of all $N$-gons with fixed volume. This result is due to a work of Bucur and Fragalà in 2014. In this note, we address the stability of their result in terms of the $L^1$ distance between sets. Furthermore, we provide a stability inequality in terms of the Hausdorff distance between the boundaries of sets in the class of polygons having uniformly bounded diameter. Finally, we show that our results are sharp, both in the exponent of decay and in the notion of distance between sets.

math.AP↗
arXiv2014

A sharp quantitative version of Hales' isoperimetric honeycomb theorem

We prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics.

math.AP↗