Search

We study the thermal properties of a composite material made up of a medium hosting an ε$$ \varepsilon $$‐periodic array of perfect thermal conductors. The thermal potentials of the two phases are coupled across the interface through a non‐standard imperfect contact transmission condition, involving the external thermal flux and a proportionality coefficient D0εα$$ {D}_0{\varepsilon}^{\alpha } $$, where α∈ℝ$$ \alpha \in \mathbb{R} $$ is a scaling parameter and D0>0$$ {D}_0>0 $$ accounts for the imperfect contact. We perform the homogenization for all the scalings α∈ℝ$$ \alpha \in \mathbb{R} $$ and we compare the resulting models with the perfect contact transmission case addressed in some of our previous studies, letting D0→0$$ {D}_0\to 0 $$, where this is meaningful. [ABSTRACT FROM AUTHOR]

An energy for first-order structured deformations in the context of periodic homogenization is obtained. This energy, defined in principle by relaxation of an initial energy of integral-type featuring contributions of bulk and interfacial terms, is proved to possess an integral representation in terms of relaxed bulk and interfacial energy densities. These energy densities, in turn, are obtained via asymptotic cell formulae defined by suitably averaging, over larger and larger cubes, the bulk and surface contributions of the initial energy. The integral representation theorem, the main result of this paper, is obtained by mixing blow-up techniques, typical in the context of structured deformations, with the averaging process underlying the theory of homogenization. [ABSTRACT FROM AUTHOR]

Diffusion in inhomogeneous materials can be described by both the Fick and Fokker–Planck diffusion equations. Here, we study a mixed Fick and Fokker–Planck diffusion problem with coefficients rapidly oscillating both in space and time. We obtain macroscopic models performing the homogenization limit by means of the unfolding technique. [ABSTRACT FROM AUTHOR]

We study, by means of the periodic unfolding technique, the homogenization of a modified bidomain model, which describes the propagation of the action potential in the cardiac electrophysiology. Such a model, allowing the presence of pathological zones in the heart, involves various geometries and non-standard transmission conditions on the interface between the healthy and the damaged part of the cardiac muscle. [ABSTRACT FROM AUTHOR]

In this paper we derive a model for heat diffusion in a composite medium in which the different components are separated by thermally active interfaces. The previous result is obtained via a concentrated capacity procedure and leads to a non-standard system of PDEs involving a Laplace-Beltrami operator acting on the interface. For such a system well-posedness is proved using contraction mapping and abstract parabolic problems theory. Finally, the exponential convergence (in time) of the solutions of our system to a steady state is proved. [ABSTRACT FROM AUTHOR]

In this paper we study a model for the heat conduction in a composite having a microscopic structure arranged in a periodic array. We obtain the macroscopic behaviour of the material and specifically the overall conductivity via an homogenization procedure, providing the equation satisfied by the effective temperature. [ABSTRACT FROM AUTHOR]

We prove a lower semicontinuity theorem for a polyconvex functional of integral form, related to maps u : Ω ⊂ ℝn → ℝm in W1,n(Ω, ℝm) with n ≥ m ≥ 2, with respect to the weak W1,p-convergence for p > m - 1, without assuming any coercivity condition. [ABSTRACT FROM AUTHOR]

We study the geodesics in a planar chessboard structure. The results for a fixed structure allow us to infer the properties of the Finsler metrics obtained, with a homogenization procedure, as limits of oscillating chessboard structures. [ABSTRACT FROM AUTHOR]

New L¹-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the lower semicontinuity in the sense of the 1-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a representation formula for the relaxed functional is also obtained. [ABSTRACT FROM AUTHOR]

We study a problem set in a finely mixed periodic medium, modelling electrical conduction in biological tissues. The unknown electric potential solves standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by a dielectric surface (the cell membranes), which exhibits both a capacitive and a nonlinear conductive behaviour. Accordingly, dynamical conditions prevail on the membranes, so that the dependence of the solution on the time variable t is not only of parametric character. As the spatial period of the medium goes to zero, the electric potential approaches in a suitable sense a homogenization limit u0, which keeps the prescribed boundary data, and solves the equation ${\rm div}\, [B^0\nabla_x u_0+\int_0^t A^1(t-\tau)\nabla_x u_0(\tau)d\tau-\mathcal{F}]=0$. This is an elliptic equation containing a term depending on the history of the gradient of u0; the matrices B0, A1 in it depend on the microstructure of the medium. More exactly, we have that, in the limit, the current is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear. The limiting equation also contains a term ℱ keeping trace of the initial data. [ABSTRACT FROM AUTHOR]

The framework of this paper is given by the mixed boundary‐value problem \[\left\{\begin{array}{@{}l@{\quad}l}\Delta u(x)=0,&x;\in \Omega,\\u(x)=0,&x;\in \Gamma _{0},\\\frac{\partial u}{\partial n}(x)=q(x),&x;\in \Gamma _{1},\end{array}\right.\] where Ω is a plane domain bounded by a regular curve composed by two arcs Γ0 and Γ1. Assuming that |Γ1|=ℇ and denoting by u[ℇ] the solution to this problem, we study some asymptotic expansions in terms of ℇ which are related to u[ℇ]. Some connections are presented among these expansions, on one hand, and the geometry of the domain Ω, on the other. In addition, a systematic way is found for computing at the boundary the Ghizzetti's integral that solves the problem. [ABSTRACT FROM AUTHOR]

Copyright of Annali dell'Universita di Ferrara: Sezione VII-Scienze Mathematiche is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)