Calculus of Variations and Geometric Measure Theory
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M. Amar - D. Andreucci - P. Bisegna - R. Gianni

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

created on 26 Nov 2002
modified by amar on 13 Dec 2004


Published Paper

Inserted: 26 nov 2002
Last Updated: 13 dec 2004

Journal: Mathematical Models and Methods In Applied Sciences
Volume: 14
Number: 9
Pages: 1261-1295
Year: 2004


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 eletric potential approaches a homogenization limit $u_{0}$, solving \begin{equation} div\Big-\sigma_{0} \nabla_{x} u_{0} - A^{0} \nabla_{x} u_{0} - \int_{0}^{t} A^{1}(t-\tau) \nabla_{x} u_{0}(x,\tau)\,d\tau +\mathcal{F}(x,t)\Big=0 \,, \end{equation} where $\sigma_0>0$ and the matrices $A^{0}$, $A^{1}$ depend on the properties of the tissue, and the vector function $\mathcal{F}$ keeps trace of the initial data of the original problem. In the limit, the current, given as the term in square brackets in the PDE above, is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear.


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