Homogenization techniques and applications to biological tissues

These notes contain the lectures of a 20/25 hours course entitled “Homogenization Techniques and Applications to Biological Tissues”, held in 2009/2010 for “Dottorato di Modelli e Metodi Matematici per la Tecnologia e la Società” at Dipartimento Me.Mo.Mat., Facoltà di Ingegneria, “Sapienza Università di Roma”. The aim of the authors is to present an introduction to homogenization techniques based on the asymptotic expansions introduced by Bensoussan-Lions-Papanicolau, jointly with an application to a particular physical problem. The basic ideas of homogenization techniques are here presented with the purpose of giving to the students a solid knowledge in this field in order to make them able to apply these techniques also in different contexts. To this aim, first we describe the usual homogenization procedure of the Dirichlet problem for a standard elliptic equation with periodic coefficients; afterwards we apply these ideas to study a physical problem, relevant both from the mathematical point of view as well as for the applications. More precisely, we present an application of homogenization techniques to study the behavior of a biological tissue subjected to an electrical current flux. Indeed, it is well known that electrical potentials are crucial for imaging techniques in medical diagnosis, in order to investigate the physical properties of biological tissues. The model which we present here is described by means of a system of elliptic equations whose solutions are coupled because of the interface conditions, since they have to satisfy the property of flux-continuity and a transmission condition of dynamic type. From the mathematical point of view, the presence of these interface conditions is a non standard problem both for the study of the well-posedness as well as for the homogenization. These notes are divided into four chapters: in Chapter 1 some preliminary notions of Functional Analysis are very briefly recalled (metric, normed and Hilbert spaces, Sobolev spaces and embedding theorems, space of bounded variation functions); these notions are necessary for a complete understanding of the course itself; in Chapter 2 we study the connections between existence of solutions to minimum problems for integral functionals and well-posedness for the Dirichlet problem for elliptic PDEs in divergence form. To this purpose we introduce the basic ideas of Convex Analysis (preliminary definitions, basic properties of lower semicontinuous and convex functions, Gˆateaux and Fr´echet derivatives) and we mention preliminary notions of Direct Methods of Calculus of Variations (coercivity, existence theorem for minimizers, applications to integral functionals). We mention also Lax-Milgram Lemma and some of its applications to linear variational PDEs. In Chapter 3 we consider the homogenization of a standard elliptic equation, introducing the technique of asymptotic expansions due to Bensoussan-Lions-Papanicolau and the energy convergence method of Tartar. Finally, in Chapter 4 we apply the previously introduced homogenization techniques to the study of a physical model governing the electrical conduction in biological tissues (Electric Impedance Tomography).