Hi everyone.

So first of all I hope you and your families are sane and healthy and I hope we can meet

as soon as possible in person.

Meanwhile we record the lectures and in particular starting from today and for the next couple

of lectures we will talk about homogenization.

So these lectures are devoted to give you a brief introduction on mathematical theory

of homogenization, which is useful to describe properly materials composed of several constituents.

In particular we will deal with periodic homogenization.

So periodic homogenization which means that we will consider heterogeneous medium with

a periodic structure because there are many applications concerning this type of materials

and in particular we will deal with boundary value problems for heterogeneous medium with

a periodic structure.

So the idea of homogenization is the following.

We have a material made of several constituents with a certain periodic structures and starting

from a microscopic analysis, so we will consider the microscopic problems if you want first,

we want to achieve a macroscopic description of the material and in order to do that we

need to consider a parameter epsilon.

So this parameter epsilon is the period or so to say is the ratio between the sides of

the sample of the medium that you are considering and the period of the material and in particular

so we want to send somehow this parameter to zero in order to achieve an average description

of the properties of the material that we are considering and this is what is called

homogenization.

Now let's start with the problem that we want to consider.

We consider a problem of diffusion or connectivity in an heterogeneous medium obtained by mixing

periodically two phases.

So we consider omega is a subset over D where D is the dimension greater or equal than one

and omega is bounded and omega is our periodic domain.

So omega is periodic domain.

I will try to draw a picture, so this is our omega, of course the picture will be better

in the lecture notes.

So this omega has some macroscopic structures, so for instance, so think that there are,

okay so let me, okay so like this and then of course we go on with this structure here

and so in each cell there are basically two phases, okay.

For instance we can represent it in the following way.

So imagine that the sides of these squares are the same.

This is epsilon and this is our period, so epsilon is the period and we consider, so

all these cells, we actually consider a rescaled unit cell, so in particular we denote the

cell by Y which is capital Y which is the interval 0, 1 to the power D, so in particular

is an interval in dimension D.

And this is the rescaled unit cell, I would say periodic cell.

Okay, now we want to, so the problem that we want to describe is a problem of conductivity

in omega.

So maybe I can, okay, so the goal, the first goal I would say is to describe, is to consider

a homogenization problem for the problem of the conductivity in the domain omega.

So we consider, so we want to consider the homogenization of a conductivity or diffusion

problem.

And now the question is how do we describe, so first of all how, so what is the conductivity

problem?

So we need the conductivity.

Of course the conductivity in omega in view of the several constituents that omega has,