Searcharxiv⌕ Search

arXiv subjects

Maria Stella Gelli

Publications and source records attributed to Maria Stella Gelli.

6 recordsLinked to original sources
arXiv2024

Variational analysis of nonlocal Dirichlet problems in periodically perforated domains

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter $\varepsilon>0$ and $Γ$-converging to local functionals defined on Sobolev spaces as $\varepsilon\to 0$. We study the asymptotic behaviour of the functionals when the order parameter is subject to Dirichlet conditions on a periodically perforated domains, given by a periodic array of small balls of radius $r_δ$ centered on a $δ$--periodic lattice, being $δ> 0$ an additional small parameter and $r_δ=o(δ)$. We highlight differences and analogies with the local case, according to the interplay between the three scales $\varepsilon$, $δ$ and $r_δ$. A fundamental tool in our analysis turns out to be a non local variant of the classical Gagliardo-Nirenberg-Sobolev inequality in Sobolev spaces which may be of independent interest and useful for other applications.

math.AP↗
arXiv2023

Mass optimization problem with convex cost

In this paper we consider a mass optimization problem in the case of scalar state function, where instead of imposing a constraint on the total mass of the competitors, we penalize the classical compliance by a convex functional defined on the space of measures. We obtain a characterization of optimal solutions to the problem through a suitable PDE. This generalizes the case considered in the literature of a linear cost and applies to the optimization of a conductor where very low and very high conductivities have both a high cost, and then the study of nonlinear models becomes relevant.

math.OC↗
arXiv2020

The role of intrinsic distances in the relaxation of $L^\infty$-functionals

We consider a supremal functional of the form $$F(u)=\mathop{\rm ess\: sup }_{x \in Ω} f(x,Du(x))$$ where $Ω\subseteq \mathbf {R}^N$ is a regular bounded open set, $u\in W^{1,\infty}(Ω)$ and $f$ is a Borel function. Assuming that the intrinsic distances $d^λ_F(x,y):= \sup \Big\{ u(x) - u(y): \, F(u)\leq λ\Big\}$ are locally equivalent to the euclidean one for every $λ>\inf_{W^{1,\infty}(Ω)} F$, we give a description of the sublevel sets of the weak$^*$-lower semicontinuous envelope of $F$ in terms of the sub-level sets of the difference quotient functionals $R_{d^λ_F}(u):=\sup_{x\not =y} \frac{u(x)-u(y)}{d^λ_F(x,y)}. $ As a consequence we prove that the relaxed functional of positive $1$-homogeneous supremal functionals coincides with $R_{d^1_F}$. Moreover, for a more general supremal functional $F$ (a priori non coercive), we prove that the sublevel sets of its relaxed functionals with respect to the weak$^*$ topology, the weak$^*$ convergence and the uniform convergence are convex. The proof of these results relies both on a deep analysis of the intrinsic distances associated to $F$ and on a careful use of variational tools such as $Γ$-convergence.

math.OC↗
arXiv2015

Asymptotic analysis of microscopic impenetrability constraints for atomistic systems

In this paper we analyze a two-dimensional discrete model of nearest-neighbour Lennard-Jones interactions under the microscopical constraint that points on a lattice triangle maintain their order. This can be understood as a microscopical non-interpenetration constraint and amounts to the positiveness of the determinant of the gradient of the piecewise-affine interpolations of the discrete displacement. Under such a constraint we examine the continuum fracture energy deriving from a discrete-to-continuum analysis at a scaling where surface energy is preponderant. We give a lower bound by an anisotropic Griffith energy. This bound is optimal if the macroscopic displacement satisfies some opening-crack conditions on the fracture site. We show that if such conditions are not satisfied then the computation of the energy due to continuum cracks may involve non-local effects necessary to bypass the positive-determinant constraint on crack surfaces and at points where more cracks meet. Even when the limit fracture energy may be described by a surface energy density, this may depend on the crack orientation both in the reference and in the deformed configuration. While these effects lead to very interesting analytical issues, they call into question the necessity of the determinant constraint for fracture problems.

math-ph↗
arXiv2013

Monotonicity formulas for obstacle problems with Lipschitz coefficients

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a Hölder continuous linear term. With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli, Weiss and Monneau.

math.AP↗