arXiv subjects

Silvia Jiménez Bolaños

Publications and source records attributed to Silvia Jiménez Bolaños.

6 recordsLinked to original sources

arXiv2026

A point-free theory of quantitative homogenization

We introduce a purely operator-theoretic framework for quantitative homogenization that bypasses the traditional reliance on large-scale spatial regularity and probabilistic assumptions. Inspired by Tartar's vision of a \emph{point-free} theory, we derive explicit norm resolvent estimates using only the algebraic structure of multiscale operators and the abstract geometry of Hilbert spaces. In this framework, the effective macroscopic dynamics and the abstract corrector emerge naturally from an orthogonal decomposition of the state space, governed algebraically by a Schur complement. To quantify the convergence rate, we introduce a frequency-splitting technique and solve a generalized Sylvester equation that controls the commutator between the differential structure and the highly oscillatory material properties. This abstract perspective unifies stationary, non-stationary, periodic, quasi-periodic, and stochastic homogenization. We demonstrate that the physical distinctions between these media--and their respective convergence rates--are entirely captured by the behavior of the spectral measures of the microscopic and macroscopic derivative operators near zero frequency.

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arXiv2025

Homogenization of a semilinear elliptic problem

We consider the homogenization of a semilinear elliptic equation where the coefficients of the second-order differential operator may be discontinuous. We establish the existence and uniqueness of the fine-scale solution, followed by an a priori estimate. The homogenized equation is derived using two-scale convergence, and a corrector result is also provided

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arXiv2024

Homogenization of high-contrast dielectric elastomer composites

This paper focuses on the homogenization of high-contrast dielectric elastomer composites, materials that deform in response to electrical stimulation. The considered heterogeneous material consisting of an ambient material with inserted particles is described by a weakly coupled system of an electrostatic equation with an elastic equation enriched with electrostriction. It is assumed that particles gradually become rigid as the fine-scale parameter approaches zero. This study demonstrates that the effective response of this system entails a homogeneous dielectric elastomer, described by a weakly coupled system of PDEs. The coefficients of the homogenized equations are dependent on various factors, including the composite's geometry, the original microstructure's periodicity, and the coefficients characterizing the initial heterogeneous material. Particularly, these coefficients are significantly influenced by the high-contrast nature of the fine-scale problem's coefficients. Consequently, as anticipated, the high-contrast coefficients of the original yield non-local effects in the homogenized response.

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arXiv2022

Homogenization for a Variational Problem with a Slip Interface Condition

Inspired by applications, we study the effect of interface slip on the effective wave propagation in poroelastic composites. The current literature on the homogenization for the poroelastic wave equations are all based on the no-slip interface condition posed on the micro-scale. However, for certain pore fluids, the no-slip conditions are known to be physically invalid. Even though there are results in a few papers regarding porous media with slip condition on the interface, they are for porous media with rigid solid matrix rather than an elastic one. For the former case, the equations for the micro-scale are posed only in the pore space and the slip on the interface involves only the fluid velocity and the fluid stress. For the latter case, both the fluid equations and the elastic equations are posed in the respective phases and the slip conditions involve the velocities on both sides of the interface, rather than just the fluid side. With this slip condition, a variational boundary value problem governing the small vibrations of a periodic mixture of an elastic solid and a slightly viscous fluid is studied in the paper. The method of two-scale convergence is used to obtain the macroscopic behavior of the solution and to identify the role played by the slip interface condition.

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arXiv2022

Homogenization of a non-linear strongly coupled model of magnetorheological fluids

This paper concerns the rigorous periodic homogenization for a non-linear strongly coupled system, which models a suspension of magnetizable rigid particles in a non-conducting carrier viscous Newtonian fluid. The fluid drags the particles, thus alters the magnetic field. Vice versa, the magnetic field acts on the particles, which in turn affect the fluid via the no-slip boundary condition. As the size of the particles approaches zero, it is shown that the suspension's behavior is governed by a generalized magnetohydrodynamic system, where the fluid is modeled by a stationary Navier-Stokes system, while the magnetic field is modeled by Maxwell equations. A corrector result from the theory of two-scale convergence allows us to obtain the limit of the product of several weakly convergent sequences, where the div-curl lemma, which is a typical tool in these types of problems, is not applicable.

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arXiv2014

Nonlinear Neutral Inclusions: Assemblages of Coated Ellipsoids

The problem of determining nonlinear neutral inclusions in (electrical or thermal)conductivity is considered. Neutral inclusions, inserted in a matrix containing a uniform applied electric field, do not disturb the field outside the inclusions. The well known Hashin coated sphere construction is an example of a neutral inclusion. In this paper, we consider the problem of constructing neutral inclusions from nonlinear materials. In particular, we discuss assemblages of coated ellipsoids.

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