Hi again, we're going to continue with another example.

This example will be a PDE based parameter identification problem.

The problem will be as follows.

We consider a domain and we call this omega.

And this domain will be a composite material of some sort.

So maybe it looks like this.

And there could be, let's say, copper here, maybe iron here, maybe wood here, maybe rubber

here, something like that.

So it's a domain that consists of different kinds of material.

So this means that it's pointwise continuous.

That's not a strong restriction.

We can drop this restriction.

You could also have something very smooth, kind of a smooth switch from copper to iron

via alloys.

This might not be piecewise constant, but it can also be continuously varying from,

let's say, copper to iron, whatever that means.

And this material corresponds to some kind of parameter.

We'll call this u.

Maybe it's minus two here, maybe it's minus one here, maybe zero here and one here.

What this u means, this is something called the thermal log conductivity.

Why this log?

Well, that means that the exponential of u is the thermal conductivity.

That's all it means.

And that makes sense.

If u is minus two, then e to the minus two is quite small.

And we know that plastic has a very low thermal conductivity, wood also, and on the other

hand, copper has a quite high thermal conductivity.

So we can think of this being a composite material of different thermal conductivities.

And we can ignore what kind of material that is.

The only interesting thing about this is the parameter u.

So this is, we can interpret u as a function from the domain of omega to r.

So each point in the domain has attached a label with its own thermal log or not log

conductivity.

So now that's a very static problem, nothing happens.

But what happens if we now apply heat to this domain?

So we first have to specify boundary conditions.

Let's say it's isolated, so perfectly thermally isolated at the lower boundary here, and it's

zero degrees Kelvin here, zero degrees Kelvin here, and zero degrees Kelvin here.

But this down so just look like, okay, maybe zero Kelvin, zero Kelvin, and so on.

And now we apply a heat source, so something like that.

So heat source.

Now what would happen if the thermal conductivity was constant on the whole domain, then well,

let's put the heat source centered.

So symmetry arguments apply.

Let's assume that the heat source is centered.

And now what happens?

Well, because of, well, this is larger than zero Kelvin.

So this is quite hot, and this is very cold at the bottom here.

And it's isolated in here.

So the heat distribution will probably look something like that.