Micol Amar

A notion of total variation depending on a metric with discontinuous coefficients

Abstract Given a function u : Ω ⊆ _ ℝ n → ℝ , we introduce a notion of total variation of u depending on a possibly discontinuous Finsler metric. We prove some integral representation results for this total variation, and we study the connections with the theory of relaxation.

Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues

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 . 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.

Homogenization limit for electrical conduction in biological tissues in the radio-frequency range

We study an evolutive model for electrical conduction in biological tissues, where the conductive intra-cellular and extracellular spaces are separated by insulating cell membranes. The mathematical scheme is an elliptic problem, with dynamical boundary conditions on the cell membranes. The problem is set in a nely mixed periodic medium. We show that the homogenization limit u0 of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation

Relaxation of quasi-convex integrals of arbitrary order

An integral representation result is given for the lower semicontinuous envelope of the functional ʃΩ f (∇ k u) dx on the space BV k (Ω:ℝ m ) of the integrable functions, whose the f -th derivative in the sense of distributions is a Radon measure with bounded total variation.

Exponential asymptotic stability for an elliptic equation with memory arising in electrical conduction in biological tissues

We study an electrical conduction problem in biological tissues in the radiofrequency range, which is governed by an elliptic equation with memory. We prove the time exponential asymptotic stability of the solution. We provide in this way both a theoretical justification to the complex elliptic problem currently used in electrical impedance tomography and additional information on the structure of the complex coefficients appearing in the elliptic equation. Our approach relies on the fact that the elliptic equation with memory is the homogenisation limit of a sequence of problems for which we prove suitable uniform estimates.

Weak lower semicontinuity for non coercive polyconvex integrals

Abstract We prove a lower semicontinuity theorem for a polyconvex functional of integral form, related to maps u : Ω ⊂ ℝ n → ℝ m in W 1,n (Ω;ℝ m ) with n ≥ m ≥ 2, with respect to the weak W 1,p -convergence for p > m – 1, without assuming any coercivity condition.

Γ-convergence for concentration problems

In this paper, we use � -convergence techniques to study the following variational problem S F e (�) := sup � e −2 ∗ � � F(u) dx : � , with 2 ∗ = 2n n−2 , andis a bounded domain of R n , n ≥ 3. We obtain a � -convergence result, on which one can easily read the usual concentration phenomena arising in critical growth problems. We extend the result to a non-homogeneous version of problem S F e (�) . Finally, a second order expansion in � -convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.

Integral representation of functionals defined on curves of W^1,p

Let I ⊂ ℝ be a bounded open interval, (I) be the family of all open subintervals of I and let p > 1. The aim of this paper is to give an integral representation result for abstract functionals F: W1,p(I;ℝn) × (I) → [0, + ∞) which are lower semicontinuous and satisfy suitable properties. In particular, we prove an integral representation theorem for the Г-limit of a sequence {Fh}h, of functionals of the form S0308210500012749_eqnU1 where each fh is a Borel function satisfying proper growth conditions

LOWER SEMICONTINUITY RESULTS FOR FREE DISCONTINUITY ENERGIES

We establish new lower semicontinuity results for energy functionals containing a very general volume term of polyconvex type and a surface term depending on the spatial variable in a discontinuous way.

On the Finsler metrics obtained as limits of chessboard structures

Abstract 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.