Searcharxiv⌕ Search

arXiv subjects

Irene Fonseca

Publications and source records attributed to Irene Fonseca.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotic analysis of higher-order perturbations of the Perona--Malik functional

The Gamma-limit of higher-order singular perturbations of the Perona-Malik functional is analyzed. The energies considered combine the critically scaled logarithmic term with a k-th order regularization designed to balance bulk and interfacial effects. A compactness result is obtained, and the Gamma-limit is identified as a free-discontinuity functional on SBV, given by the sum of the Dirichlet energy and a surface term proportional to the jump amplitude to the power 1/k. The surface density is characterized through a one-dimensional optimal-profile problem with homogeneous boundary conditions on derivatives up to order k-1. As a consequence, the limit of the same energies at a different scaling is determined. That scaling had been previously studied in the second-order case to address the so-called staircasing phenomenon.

math.AP↗
arXiv2025

$\Gamma$-convergence and stochastic homogenization for functionals in the $\mathcal{A}$-free setting

We obtain a compactness result for $\Gamma$-convergence of integral functionals defined on $\mathcal{A}$-free vector fields. This is used to study homogenization problems for these functionals without periodicity assumptions. More precisely, we prove that the homogenized integrand can be obtained by taking limits of minimum values of suitable minimization problems on large cubes, when the side length of these cubes tends to $+\infty$, assuming that these limit values do not depend on the center of the cube. Under the usual stochastic periodicity assumptions, this result is then used to solve the stochastic homogenization problem by means of the subadditive ergodic theorem.

math.AP↗
arXiv2025

Monotonicity of the jump set and jump amplitudes in one-dimensional TV denoising

We revisit the classical problem of denoising a one-dimensional scalar-valued function by minimizing the sum of an $L^2$ fidelity term and the total variation, scaled by a regularization parameter. This study focuses on proving that the jump set of solutions, corresponding to discontinuities or edges, as well as the amplitude of the jumps are nonincreasing as the regularization parameter increases. Compared with previous works, our results apply to a strictly larger class of input functions, extending beyond the traditional setting of functions of bounded variation to any input in $L^\infty$ with left and right approximate limits everywhere. The proof leverages competitor constructions and convexity properties of the taut string problem, a well-known equivalent formulation of the TV model. This monotonicity property reflects that the extent to which geometric and topological features of the original signal are preserved is consistent with the amount of smoothing desired when formulating the denoising method.

math.OC↗
arXiv2025

Second-Order $Γ$-Limit for the Cahn-Hilliard Functional with Dirichlet Boundary Conditions, II

This paper continues the study of the asymptotic development of order 2 by $Γ$ -convergence of the Cahn-Hilliard functional with Dirichlet boundary conditions initiated in [8]. While in the first paper, the Dirichlet data are assumed to be well separated from one of the two wells, here this is no longer the case. In the case where there are no interfaces, it is shown that there is a transition layer near the boundary of the domain.

math.AP↗
arXiv2024

Motion of Islands of Elastic Thin Films in the Dewetting Regime

This paper addresses a two-dimensional sharp interface variational model for solid-state dewetting of thin films with surface energies, introduced by Wang, Jiang, Bao, and Srolovitz in \cite{jiang2016solid}. Using the $H^{-1}$-gradient flow structure of the evolution law, short-time existence for a surface diffusion evolution equation with curvature regularization is established in the context of epitaxially strained two-dimensional films. The main novelty, as compared to the study of the wetting regime, is the presence of moving contact lines.

math.AP↗
arXiv2024

Dyadic partition-based training schemes for TV/TGV denoising

Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the best-known methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image pairs. In this work, we consider such methods with space-dependent parameters which are piecewise constant on dyadic grids, with the grid itself being part of the minimization. We prove existence of minimizers for fixed discontinuous parameters under mild assumptions on the data, which lead to existence of finite optimal partitions. We further establish that these assumptions are equivalent to the commonly used box constraints on the parameters. On the numerical side, we consider a simple subdivision scheme for optimal partitions built on top of any other bilevel optimization method for scalar parameters, and demonstrate its improved performance on some representative test images when compared with constant optimized parameters.

math.AP↗
arXiv2023

$Γ$-Convergence of an Ambrosio-Tortorelli approximation scheme for image segmentation

Given an image $u_0$, the aim of minimising the Mumford-Shah functional is to find a decomposition of the image domain into sub-domains and a piecewise smooth approximation $u$ of $u_0$ such that $u$ varies smoothly within each sub-domain. Since the Mumford-Shah functional is highly non-smooth, regularizations such as the Ambrosio-Tortorelli approximation can be considered which is one of the most computationally efficient approximations of the Mumford-Shah functional for image segmentation. While very impressive numerical results have been achieved in a large range of applications when minimising the functional, no analytical results are currently available for minimizers of the functional in the piecewise smooth setting, and this is the goal of this work. Our main result is the $Γ$-convergence of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional for piecewise smooth approximations. This requires the introduction of an appropriate function space. As a consequence of our $Γ$-convergence result, we can infer the convergence of minimizers of the respective functionals.

math.OC↗
arXiv2023

Homogenization and Phase Separation with Fixed Wells -- The Supercritical Case

A variational model for the interaction between homogenization and phase separation is considered in the regime where the former happens at a finer scale than the latter. The first order $Γ-$limit is proven to exhibit a separation of scales which has been previously conjectured in a paper by Bradies and Zeppieri and in the PhD thesis of Hagerty.

math.AP↗
arXiv2023

Vector Field Models for Nematic Disclinations

In this paper, a model for defects that was introduced in \cite{ZANV} is studied. In the literature, the setting of most models for defects is the function space SBV (special bounded variation functions) (see, e.g., \cite{ContiGarroni, GoldmanSerfaty}). However, this model regularizes the director field to be in a Sobolev space by adding a second field to incorporate the defect. A relaxation result in the case of fixed parameters is proven along with some partial compactness results.

math.AP↗
arXiv2022

Homogenization and phase separation with space dependent wells -- The subcritical case

A variational model for the interaction between homogenization and phase separation is considered. The focus is on the regime where the latter happens at a smaller scale than the former, and when the wells of the double well potential are allowed to move and to have discontinuities. The zeroth and first order $Γ$-limits are identified. The topology considered for the latter is that of two-scale, since it encodes more information on the asymptotic local microstructure. In particular, when the wells are non constant, the first order $Γ$-limit describes the contribution of microscopic phase separation, also in situations where there is no macroscopic phase separation. As a corollary, the minimum of the mass constrained minimization problem is characterized, and it is shown to depend on whether or not the wells are discontinuous. In the process of proving these results, the theory of inhomogeneous Modica Mortola functionals is strengthened.

math.AP↗
arXiv2021

Anisotropic Surface Tensions for Phase Transitions in Periodic Media

This paper establishes bounds on the homogenized surface tension for a heterogeneous Allen-Cahn energy functional in a periodic medium. The approach is based on relating the homogenized energy to a purely geometric variational problem involving the large scale behaviour of the signed distance function to a hyperplane in periodic media. Motivated by this, a homogenization result for the signed distance function to a hyperplane in both periodic and almost periodic media is proven.

math.AP↗
arXiv2020

A homogenization result in the gradient theory of phase transitions

A variational model in the context of the gradient theory for fluid-fluid phase transitions with small scale heterogeneities is studied. In particular, the case where the scale $\varepsilon$ of the small homogeneities is of the same order of the scale governing the phase transition is considered. The interaction between homogenization and the phase transitions process will lead, in the limit as $\varepsilon\to0$, to an anisotropic interfacial energy.

math.AP↗
arXiv2020

Homogenization of quasi-crystalline functionals via two-scale-cut-and-project convergence

We consider a homogenization problem associated with quasi-crystalline multiple integrals of the form \begin{equation*} \begin{aligned} u_\varepsilon\in L^p(Ω;\mathbb{R}^d) \mapsto \int_Ωf_R\Big(x,\frac{x}{\varepsilon}, u_\varepsilon(x)\Big)\, dx, \end{aligned} \end{equation*} where $u_\varepsilon$ is subject to constant-coefficient linear partial differential constraints. The quasi-crystalline structure of the underlying composite is encoded in the dependence on the second variable of the Lagrangian, $f_R$, and is modeled via the cut-and-project scheme that interprets the heterogeneous microstructure to be homogenized as an irrational subspace of a higher-dimensional space. A key step in our analysis is the characterization of the quasi-crystalline two-scale limits of sequences of the vector fields $u_\varepsilon$ that are in the kernel of a given constant-coefficient linear partial differential operator, $\mathcal{A}$, that is, $\mathcal{A} u _\varepsilon =0$. Our results provide a generalization of related ones in the literature concerning the ${\rm \mathcal{A} =curl } $ case to more general differential operators $\mathcal{A}$ with constant coefficients, and without coercivity assumptions on the Lagrangian $f_R$.

math.AP↗
arXiv2020

On the motion of curved dislocations in three dimensions: Simplified linearized elasticity

It is shown that in core-radius cutoff regularized simplified elasticity (where the elastic energy depends quadratically on the full displacement gradient rather than its symmetrized version), the force on a dislocation curve by the negative gradient of the elastic energy asymptotically approaches the mean curvature of the curve as the cutoff radius converges to zero. Rigorous error bounds in Hölder spaces are provided. As an application, convergence of dislocations moving by the gradient flow of the elastic energy to dislocations moving by the gradient flow of the arclength functional, when the motion law is given by an $H^1$-type dissipation, and convergence to curve shortening flow in co-dimension $2$ for the usual $L^2$-dissipation is established. In the second scenario, existence and regularity are assumed while the $H^1$-gradient flow is treated in full generality (for short time). The methods developed here are a blueprint for the more physical setting of linearized isotropic elasticity.

math.AP↗
arXiv2018

Piecewise constant reconstruction of damaged color images

A variational model for reconstruction of damaged color images is studied, in particular in the case where only finitely many colors are admissible for the reconstructed image. An existence result and regularity properties of minimizers are presented.

math.OC↗
arXiv2017

A second order minimality condition for a free-boundary problem

The goal of this paper is to derive in the two-dimensional case necessary and sufficient minimality conditions in terms of the second variation for the functional \[ v\mapsto\int_Ω\big(|\nabla v|^{2}+χ_{\{v>0\}}Q^{2} \big)\,dx, \] introduced in a classical paper of Alt and Caffarelli. For a special choice of $Q$ this includes water waves. The second variation is obtained by computing the second derivative of the functional along suitable variations of the free boundary. It is proved that the strict positivity of the second variation gives a sufficient condition for local minimality. Also, it is shown that smooth critical points are local minimizers in a small tubular neighborhood of the free-boundary.

math.AP↗