arXiv subjects

Riccardo Scala

Publications and source records attributed to Riccardo Scala.

At least 19 recordsLinked to original sources

arXiv2026

Explicit formulas of strict BV relaxed energies for polyconvex functionals with linear growth

We consider polyconvex functionals defined on vector valued functions, the model case being the graph area functional, and we analyze the relaxed energy with respect to the strict convergence in $BV$. In several cases where the relaxed area is known, by exploiting a continuity result by Reshetnyak we are able to find an explicit formula for wide classes of integrands with linear growth in the minors of the gradient. We preliminarily extend to the $BV$ setting a continuity property observed by Acerbi--Dal Maso in the Sobolev case. Finally, partial results concerning the relaxation of the vortex map with respect to the $L^1$ convergence are obtained.

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arXiv2026

Concentration effects and $\Gamma$-limit for the elastica functional for open and closed curves

We study the $\Gamma$-convergence of a class of elastica-type energies defined on immersed planar curves and depending on a small positive parameter $\epsilon$. As $\epsilon\to 0^+$, sequences with equibounded energy develop concentration phenomena in the curvature, leading to the emergence of singularities described by atomic measures. This naturally gives rise to a limiting framework in terms of pointed curves, consisting of a curve together with a measure encoding curvature concentration. We characterize the first-order $\Gamma$-limit in two settings: for immersed open curves with fixed endpoints and boundary conditions on the tangents, and for immersed closed curves of prescribed length. In both cases, the limiting energy depends only on the number of concentration points and is expressed as a sum of contributions, each given by an integer multiple of $2\pi$. A key feature of the problem is that the rescaled energies exhibit a structure closely related to one-dimensional Modica--Mortola type functionals.

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arXiv2026

$\Gamma$-convergence of free discontinuity problems for circle-valued maps in the linear regime

We investigate the $\Gamma$-convergence of Ambrosio-Tortorelli type-functionals for circle valued functions, in the case of volume terms with linear growth. We show the emergence of a non-local $\Gamma$-limit, which is due to the topological structure of the target space, and discuss compactness of minimal liftings. Our results extend the analysis of a previous work on the quadratic case.

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arXiv2026

A fourth-order regularization of the curvature flow of immersed plane curves with Dirichlet boundary conditions

We consider a fourth-order regularization of the curvature flow for an immersed plane curve with fixed boundary, using an elastica-type functional depending on a small positive parameter $\varepsilon$. We show that the approximating flow smoothly converges, as $\varepsilon \to 0^+$, to the curvature flow of the curve with Dirichlet boundary conditions for all times before the first singularity of the limit flow.

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arXiv2025

On the relaxation of polyconvex functionals with linear growth under strict convergence in $BV$

We consider the relaxation of polyconvex functionals with linear growth with respect to the strict convergence in the space of functions of bounded variation. These functionals appears as relaxation of $F(u,\Omega):=\int_\Omega f(\nabla u)dx$, where $u:\Omega\rightarrow \mathbb R^m$, and $f$ is polyconvex. In constrast with the case of relaxation with respect to the standard $L^1$-convergence, in the case that $\Omega$ is $2$-dimensional, we prove that the sets map $A\mapsto F(u,A)$ for $A$ open, is, for every $u\in BV(\Omega;\mathbb R^m)$, $m\geq1$, the restriction of a Borel measure. This is not true in the case $\Omega\subset\mathbb R^n$, with $n\geq3$. Using the integral representation formula for a special class of functions, we also show the presence of Cartesian maps whose relaxed area functional with respect to the $L^1$-convergence is strictly larger than the area of its graph.

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arXiv2025

On jump minimizing liftings for $\mathbb S^1$-valued maps and connections with Ambrosio-Tortorelli-type $\Gamma$-limits

This paper is concerned with the $\Gamma$-limits of Ambrosio-Tortorelli-type functionals, for maps $u$ defined on an open bounded set $\Omega\subset\mathbb R^n$ and taking values in the unit circle $\mathbb S^1\subset\mathbb R^2$. 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.

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arXiv2024

Relaxation of the area of the vortex map: a non-parametric Plateau problem for a catenoid containing a segment

Motivated by the study of the non-parametric area $\mathcal A$ of the graph of the vortex map $u$ (a two-codimensional singular surface in $\mathbb R^4$) over the disc $\Omega \subset \mathbb R^2$ of radius $l$, we perform a careful analysis of the singular part of the relaxation of $\mathcal A$ computed at $u$. The precise description is given in terms of a area-minimizing surface in a vertical copy of $\mathbb R^3 \subset \mathbb R^4$, which is a sort of ``catenoid'' containing a segment corresponding to a radius of $\Omega$. The problem involves an area-minimization with a free boundary part; several boundary regularity properties of the minimizer are inspected.

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arXiv2024

The $L^1$-relaxed area of the graph of the vortex map: optimal upper bound

We compute an upper bound for the value of the $L^1$-relaxed area of the graph of the vortex map $u : B_l(0)\subset \mathbb R^2 \to \mathbb R^2$, $u(x):= x/\vert x\vert$, $x \neq 0$, for all values of $l>0$. Together with a previously proven lower bound, this upper bound turns out to be optimal. Interestingly, for the radius $l$ in a certain range, in particular $l$ not too large, a Plateau-type problem, having as solution a sort of catenoid constrained to contain a segment, has to be solved.

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arXiv2024

Approximation of topological singularities through free discontinuity functionals: the critical and super-critical regimes

We further investigate the properties of an approach to topological singularities through free discontinuity functionals of Mumford-Shah type proposed in \cite{DLSVG}. We prove the variational equivalence between such energies, Ginzburg-Landau, and Core-Radius for anti-plane screw dislocations energies in dimension two, in the relevant energetic regimes $|\log \varepsilon|^a$, $a\geq 1$, where $\varepsilon$ denotes the linear size of the process zone near the defects. Further, we remove the \emph{a priori} restrictive assumptions that the approximating order parameters have compact jump set. This is obtained by proving a new density result for $\mathbb S^1$-valued $SBV^p$ functions, approximated through functions with essentially closed jump set, in the strong $BV$ norm.

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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

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arXiv2023

Upper bounds for the relaxed area of $\mathbb S^1$-valued Sobolev maps and its countably subadditive interior envelope

Given a bounded open connected Lipschitz set $Ω\subset \mathbb R^2$, we show that the relaxed Cartesian area functional $\overline{\mathcal A}(u,Ω)$ of a map $u\in W^{1,1}(Ω;\mathbb S^1)$ is finite, and provide a useful upper bound for its value. Using this estimate, we prove a modified version of a De Giorgi conjecture [17] adapted to $W^{1,1}(Ω;\mathbb S^1)$, on the largest countably subadditive set function $\overline {\overline{\mathcal A}}(u, \cdot)$ smaller than or equal to $\overline{\mathcal A}(u,\cdot)$.

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arXiv2023

Relaxed area of graphs of piecewise Lipschitz maps in the strict $BV$-convergence

We compute the relaxed Cartesian area in the strict $BV$-convergence on a class of piecewise Lipschitz maps from the plane to the plane, having jump made of several curves allowed to meet at a finite number of junction points. We show that the domain of this relaxed area is strictly contained in the domain of the classical $L^1$-relaxed area.

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arXiv2023

The relaxed area of $\mathcal{S}^1$-valued singular maps in the strict $BV$-convergence

Given a bounded open set $Ω\subset \mathbb{R}^2$, we study the relaxation of the nonparametric area functional in the strict topology in $BV(Ω;\mathbb{R}^2)$, and compute it for vortex-type maps, and more generally for maps in $W^{1,1}(Ω;\mathcal{S}^1)$ having a finite number of topological singularities. We also extend the analysis to some specific piecewise constant maps in $BV(Ω;\mathcal{S}^1)$, including the symmetric triple junction map.

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arXiv2022

A new approach to topological singularities via a weak notion of Jacobian for functions of bounded variation

We introduce a weak notion of $2\times 2$-minors of gradients of a suitable subclass of $BV$ functions. In the case of maps in $BV(\mathbb{R}^2;\mathbb{R}^2)$ such a notion extends the standard definition of Jacobian determinant to non-Sobolev maps. We use this distributional Jacobian to prove a compactness and $Γ$-convergence result for a new model describing the emergence of topological singularities in two dimensions, in the spirit of Ginzburg-Landau and core-radius approaches. Within our framework, the order parameter is an $SBV$ map $u$ taking values in $\mathbb{S}^1$ and the energy is made by the sum of the squared $L^2$ norm of $\nabla u$ and of the length of (the closure of) the jump set of $u$ multiplied by $\frac 1 \varepsilon$. Here, $\varepsilon$ is a length-scale parameter. We show that, in the $|\log\varepsilon|$ regime, the Jacobian distributions converge, as $\varepsilon\to 0^+$, to a finite sum $μ$ of Dirac deltas with weights multiple of $π$, and that the corresponding effective energy is given by the total variation of $μ$.

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arXiv2022

A non-parametric Plateau problem with partial free boundary

We consider a Plateau problem in codimension $1$ in the non-parametric setting. A Dirichlet boundary datum is given only on part of the boundary $\partial Ω$ of a bounded convex domain $Ω\subset\mathbb{R}^2$. Where the Dirichlet datum is not prescribed, we allow a free contact with the horizontal plane. We show existence of a solution, and prove regularity for the corresponding minimal surface. Finally we compare these solutions with the classical minimal surfaces of Meeks and Yau, and show that they are equivalent when the Dirichlet boundary datum is assigned in at most $2$ disjoint arcs of $\partial Ω$.

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arXiv2021

The $L^1$-relaxed area of the graph of the vortex map

We compute the value of the $L^1$-relaxed area of the graph of the map $u : B_l(0)\subset \mathbb R^2 \mapsto \mathbb R^2$, $u(x):= x/\vert x\vert$, $x \neq 0$, for all values of $l>0$. Interestingly, for $l$ in a certain range, in particular $l$ not too large, a Plateau-type problem, having as solution a sort of catenoid constrained to contain a segment, has to be solved.

math.AP↗

arXiv2020

An optimization problem in thermal insulation with Robin boundary conditions

We study thermal insulating of a bounded body $Ω\subset \mathbb{R}^n$. Under a prescribed heat source $f\geq 0$, we consider a model of heat transfer between $Ω$ and the environment determined by convection; this corresponds, before insulation, to Robin boundary conditions. The body is then surrounded by a layer of insulating material of thickness of size $\varepsilon>0$, and whose conductivity is also proportional to $\varepsilon$. This corresponds to the case of a small amount of insulating material, with excellent insulating properties. We then compute the $Γ$-limit of the energy functional $F_\varepsilon$ and prove that this is a functional $F$ whose minimizers still satisfy an elliptic PDEs system with a non uniform Robin boundary condition depending on the distribution of insulating layer around $Ω$. In a second step we study the maximization of heat content (which measures the goodness of the insulation) among all the possible distributions of insulating material with fixed mass, and prove an optimal upper bound in terms of geometric properties. Eventually we prove a conjecture which states that the ball surrounded by a uniform distribution of insulating material maximizes the heat content.

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arXiv2020

Existence, energy identity and higher time regularity of solutions to a dynamic visco-elastic cohesive interface model

We study the dynamics of visco-elastic materials coupled by a common cohesive interface (or, equivalently, {two single domains separated by} a prescribed cohesive crack) in the anti-plane setting. We consider a general class of traction-separation laws featuring an activation threshold on the normal stress, softening and elastic unloading. In strong form, the evolution is described by a system of PDEs coupling momentum balance (in the bulk) with transmission and Karush-Kuhn-Tucker conditions (on the interface). We provide a detailed analysis of the system. We first prove existence of a weak solution, employing a time discrete approach and a regularization of the initial data. Then, we prove our main results: the energy identity and the existence of { solutions} with acceleration in $L^\infty (0,T; L^2)$.

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