Okay, today we go on with homogenization, with periodic homogenization.

So this is the lecture three, which is actually lecture nine on studon.

This is the third lecture on periodic homogenization, and if you remember last time we mentioned two methods to rigorously justify the process of homogenization from a mathematical point of view.

And we mentioned basically two methods to rigorously justify homogenization,

which are, so the first was the oscillating test function method.

And the second one is actually a method that we're going to see today, and this is the so-called two-scale convergence method.

So two-scale convergence. I just remind you that so this method consists in choosing a particular type of test function,

actually a sequence of oscillating test function, which means basically test function similar to the ansatz that we have in homogenization,

so for the homogenization of our diffusion problem.

While today we see a method which actually seems to be more natural in the context and is basically based on a new concept of convergence, which is two-scale convergence.

This method was first developed by Gutzeng, I hope the pronunciation is right, in 1989, and then by Aller in 92, if I remember well.

Today I will actually explain to you this method, and I refer to these two references for further details on technical details that we're going to see today.

Okay, so let's start with describing this method. In particular, so we first give a definition.

So definition one, what does it mean, two-scale convergence?

So a sequence of functions u epsilon in L2 of omega.

Omega is always our, let me remind you, is a subset over D bounded, D is greater or equal than one, and is basically a periodic domain, I would say.

So which means basically is made of periodic microstructures.

Let me also remind you that Y, capital Y is our rescaled, a unit rescaled periodic cell. Epsilon is our period.

Now, let me just remind you that actually, differently from the first method, actually, the two-scale convergence method only works for periodic domains.

Let me write here, only for periodic problems.

Okay, so differently from the first method.

Anyhow, so what does it mean, two-scale convergence? A sequence of functions u epsilon in L2 of omega.

Two-scale convergence to a limit.

U naught in L2 omega times capital Y.

If for any test function, for any test function phi, depending on X, Y, which is C infinity and compactly supported from omega to

the space of C infinity, compactly supported and periodic function Y.

If for any test function of this type, we have the limit of epsilon equals to zero of the integral over omega of u epsilon X

tends to be phi X, X over epsilon, dx is equal to the integral over omega, integral over capital Y of u naught,

X, Y, phi, X, Y, dx, dy.

Well, of course, here, in principle, we should have one over the measure of the set Y.

But in our case, I remind you that, okay, so this is the square of area actually one. So then the measure of Y is one.

So that's why here we do not have the measure. So we don't divide by the measure of the set that we consider.

All right. So this is actually the definition of two-scale convergence. It's somehow a sort of, it's similar to weak convergence,

if you see, because we can, of course, we can, so somehow in a general case, we can treat, so this is nothing but the average of u naught

in some sense. We will see in a few seconds in how. And yeah, and actually the main motivation under this new definition of convergence

is actually the following compactness theorem. I would say discovered by Goodson and Aller.

So theorem, actually the right, so the right reference for this theorem is actually theorem 1.2

in a paper of Aller in 1992. You can find the right references in the lecture notes. I will give you the right reference there.

And in particular, if you want to have more details about this theorem, you can go directly to theorem 1.2, if I'm not wrong,

in the paper that I will give you in references. Now, so what does it say, this theorem? So it says that from each

bounded sequence u epsilon in L2 omega, so there exists a subsequence u epsilon k.

And a limit u naught in L2 of omega times capital Y such that

such that u epsilon k to scale converges to u naught.

So as you can see here, so this is a, this kind of generalization of the compactness theorem in L2,

because we know that from each bounded sequence in L2 we can extract a subsequence which is weakly converging to L2.

And in this case, we actually have, this is also true in, so in particular, we can always, so from each bounded sequence in L2,

we can always find a subsequence and a limit such that we have two scale convergence to this limit.

So, and in particular, the main motivation behind the new definition of two scale convergence actually comes from this compactness theorem.

As I said, so this theorem, so you can see all the details of the proof in theorem 1.2 in this paper of Aller.

Actually, it's a Siam, so I can just give you more details on this paper.