Marco Caroccia
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View article: Rigidity and functional properties of $\mathrm{BD}_{dev}(Ω)$
Rigidity and functional properties of $\mathrm{BD}_{dev}(Ω)$ Open
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 rig…
View article: Iterative blow-ups for maps with bounded $\mathcal{A}$-variation: a refinement, with application to $\mathrm{BD}$ and $\mathrm{BV}$
Iterative blow-ups for maps with bounded $\mathcal{A}$-variation: a refinement, with application to $\mathrm{BD}$ and $\mathrm{BV}$ Open
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 seq…
View article: On the emergence of almost-honeycomb structures in low-energy planar clusters
On the emergence of almost-honeycomb structures in low-energy planar clusters Open
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 …
View article: On the singular planar Plateau problem
On the singular planar Plateau problem Open
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.…
View article: Asymptotic Behavior of the Dirichlet Energy on Poisson Point Clouds
Asymptotic Behavior of the Dirichlet Energy on Poisson Point Clouds Open
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 D…
View article: Isoperimetric sets and $p$-Cheeger sets are in bijection
Isoperimetric sets and $p$-Cheeger sets are in bijection Open
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 …
View article: A Compactness Theorem for functions on Poisson point clouds
A Compactness Theorem for functions on Poisson point clouds Open
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-…
View article: Asymptotic behavior of the Dirichlet energy on Poisson point clouds
Asymptotic behavior of the Dirichlet energy on Poisson point clouds Open
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 D…
View article: Dimensional lower bounds for contact surfaces of Cheeger sets
Dimensional lower bounds for contact surfaces of Cheeger sets Open
We carry out an analysis of the size of the contact surface between a Cheeger set E and its ambient space Ω⊂Rd. By providing bounds on the Hausdorff dimension of the contact surface ∂E∩∂Ω, we show a fruitful interplay between this size its…
View article: Contact surface of Cheeger sets
Contact surface of Cheeger sets Open
We carry on an analysis of the size of the contact surface of a Cheeger set $E$ with the boundary of its ambient space $\Omega$. We show that this size is strongly related to the regularity of $\partial \Omega$ by providing bounds on the H…
View article: On the Integral Representation of Variational Functionals on $BD$
On the Integral Representation of Variational Functionals on $BD$ Open
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…
View article: On the integral representation of variational functionals on BD
On the integral representation of variational functionals on BD Open
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…
View article: Damage-driven fracture with low-order potentials: asymptotic behavior, existence and applications
Damage-driven fracture with low-order potentials: asymptotic behavior, existence and applications Open
We study the Γ-convergence of damage to fracture energy functionals in the presence of low-order nonlinear potentials that allows us to model physical phenomena such as fluid-driven fracturing, plastic slip, and the satisfaction of kinemat…
View article: Damage-driven fracture with low-order potentials: asymptotic behavior,\n existence and applications
Damage-driven fracture with low-order potentials: asymptotic behavior,\n existence and applications Open
We study the $\\Gamma$-convergence of damage to fracture energy functionals in\nthe presence of low-order nonlinear potentials that allows us to model physical\nphenomena such as fluid-driven fracturing, plastic slip, and the satisfaction\…
View article: Damage-driven fracture with low-order potentials: asymptotic behavior and applications
Damage-driven fracture with low-order potentials: asymptotic behavior and applications Open
We study the $\Gamma$-convergence of damage to fracture energy functionals in the presence of low-order nonlinear potentials that allow us to model physical phenomena such as of fluid-driven fracturing, plastic slip, and the satisfaction o…
View article: The Cheeger N-problem in terms of BV functions
The Cheeger N-problem in terms of BV functions Open
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- …
View article: On the isoperimetric properties of Planar N-clusters
On the isoperimetric properties of Planar N-clusters Open
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 isop…
View article: A sharp quantitative version of Hales' isoperimetric honeycomb theorem
A sharp quantitative version of Hales' isoperimetric honeycomb theorem Open
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…