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Valerio Pagliari

Publications and source records attributed to Valerio Pagliari.

12 recordsLinked to original sources
arXiv2025

On the existence of extensions for manifold-valued Sobolev maps on perforated domains

Motivated by manifold-constrained homogenization problems, we construct suitable extensions for Sobolev functions defined on a perforated domain and taking values in a compact, connected $C^2$-manifold without boundary. The proof combines a by now classical extension result for the unconstrained case with a retraction argument that heavily relies on the topological properties of the manifold. With the ultimate goal of providing necessary conditions for the existence of extensions for Sobolev maps between manifolds, we additionally investigate the relationship between this problem and the surjectivity of the trace operator for such functions.

math.AP↗
arXiv2024

Homogenization of high-contrast media in finite-strain elastoplasticity

This work is devoted to the analysis of the interplay between internal variables and high-contrast microstructure in inelastic solids. As a concrete case-study, by means of variational techniques, we derive a macroscopic description for an elastoplastic medium. Specifically, we consider a composite obtained by filling the voids of a periodically perforated stiff matrix by soft inclusions. We study the $Γ$-convergence of the related energy functionals as the periodicity tends to zero. The main challenge is posed by the lack of coercivity brought about by the degeneracy of the material properties in the soft part. We prove that the $Γ$-limit, which we compute with respect to a suitable notion of convergence, is the sum of the contributions resulting from each of the two components separately. Eventually, convergence of the energy minimizing configurations is obtained.

math.AP↗
arXiv2024

A homogenization result in finite plasticity

We carry out a variational study for integral functionals that model the stored energy of a heterogeneous material governed by finite-strain elastoplasticity with hardening. Assuming that the composite has a periodic microscopic structure, we establish the $Γ$-convergence of the energies in the limiting of vanishing periodicity. The constraint that plastic deformations belong to $\mathsf{SL}(3)$ poses the biggest hurdle to the analysis, and we address it by regarding $\mathsf{SL}(3)$ as a Finsler manifold.

math.AP↗
arXiv2023

Sharp conditions for the validity of the Bourgain-Brezis-Mironescu formula

Following the seminal paper by Bourgain, Brezis and Mironescu, we focus on the asymptotic behavior of some nonlocal functionals that, for each $u\in L^2(\mathbb{R}^N)$, are defined as the double integrals of weighted, squared difference quotients of $u$. Given a family of weights $\{ρ_ε\}$, $ε\in (0,1)$, we devise sufficient and necessary conditions on $\{ρ_ε\}$ for the associated nonlocal functionals to converge as $ε\to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.

math.AP↗
arXiv2021

Bilevel training schemes in imaging for total-variation-type functionals with convex integrands

In the context of image processing, given a $k$-th order, homogeneous and linear differential operator with constant coefficients, we study a class of variational problems whose regularizing terms depend on the operator. Precisely, the regularizers are integrals of spatially inhomogeneous integrands with convex dependence on the differential operator applied to the image function. The setting is made rigorous by means of the theory of Radon measures and of suitable function spaces modeled on $BV$. We prove the lower semicontinuity of the functionals at stake and existence of minimizers for the corresponding variational problems. Then, we embed the latter into a bilevel scheme in order to automatically compute the space-dependent regularization parameters, thus allowing for good flexibility and preservation of details in the reconstructed image. We establish existence of optima for the scheme and we finally substantiate its feasibility by numerical examples in image denoising. The cases that we treat are Huber versions of the first and second order total variation with both the Huber and the regularization parameter being spatially dependent. Notably the spatially dependent version of second order total variation produces high quality reconstructions when compared to regularizations of similar type, and the introduction of the spatially dependent Huber parameter leads to a further enhancement of the image details.

math.NA↗
arXiv2021

Homogenization of high-contrast composites under differential constraints

We derive, by means of variational techniques, a limiting description for a class of integral functionals under linear differential constraints. The functionals are designed to encode the energy of a high-contrast composite, that is, a heterogeneous material which, at a microscopic level, consists of a periodically perforated matrix whose cavities are occupied by a filling with very different physical properties. Our main result provides a $Γ$-convergence analysis as the periodicity tends to zero, and shows that the variational limit of the functionals at stake is the sum of two contributions, one resulting from the energy stored in the matrix and the other from the energy stored in the inclusions. As a consequence of the underlying high-contrast structure, the study is faced with a lack of coercivity with respect to the standard topologies in $L^p$, which we tackle by means of two-scale convergence techniques. In order to handle the differential constraints, instead, we establish new results about the existence of potentials and of constraint-preserving extension operators for linear, $k$-th order, homogeneous differential operators with constant coefficients and constant rank.

math.AP↗
arXiv2021

Convergence of nonlocal geometric flows to anisotropic mean curvature motion

We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result is achieved by combining a compactness argument and a set-theoretic approach related to the theory of De Giorgi's barriers for evolution equations.

math.AP↗
arXiv2020

Asymptotic behaviour of rescaled nonlocal functionals and evolutions

Taking up a variational viewpoint, we present some nonlocal-to-local asymptotic results for various kinds of integral functionals. The content of the thesis comprises the contributions first appeared in some research papers in collaboration with J. Berendsen, A. Cesaroni, A. Chambolle, and M. Novaga.

math.AP↗
arXiv2020

On the convergence rate of some nonlocal energies

We study the rate of convergence of some nonlocal functionals recently considered by Bourgain, Brezis and Mironescu. In particular we establish the $Γ$-convergence of the corresponding rate functionals, suitably rescaled, to a limit functional of second order.

math.AP↗
arXiv2019

Halfspaces minimise nonlocal perimeter: a proof via calibrations

We consider a nonlocal functional $J_K$ that may be regarded as a nonlocal version of the total variation. More precisely, for any measurable function $u\colon \mathbb{R}^d \to \mathbb{R}$, we define $J_K(u)$ as the integral of weighted differences of $u$. The weight is encoded by a positive kernel $K$, possibly singular in the origin. We study the minimisation of this energy under prescribed boundary conditions, and we introduce a notion of calibration suited for this nonlocal problem. Our first result shows that the existence of a calibration is a sufficient condition for a function to be a minimiser. As an application of this criterion, we prove that halfspaces are the unique minimisers of $J_K$ in a ball, provided they are admissible competitors. Finally, we outline how to exploit the optimality of hyperplanes to recover a $Γ$-convergence result concerning the scaling limit of $J_K$.

math.OC↗
arXiv2018

On the asymptotic behaviour of nonlocal perimeters

We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the paper, we show that these functionals are indeed perimeters in a generalised sense and we establish existence of minimisers for the corresponding Plateau problem. Also, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe flat boundary conditions. A Gamma-convergence result is discussed in the second part of the work. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel that has faster-than-L1 decay at infinity and we show that the Gamma-limit is the classical perimeter, up to a multiplicative constant that we compute explicitly.

math.AP↗