Let us conclude this important inequality.

So this is something quite remarkable, right?

Because we have, you know, we are considering the wave equation in any domain.

So as you know, there is an important wave process ongoing that we can understand either

by the rates of geometric optics, or if one prefers as vibrations described in Fourier

series.

And it turns out that, you know, by a rather simple proof, simply using multipliers and

integration by parts, we are able to show that boundary observations, right, are able to

recover the energy inside, right?

And of course, this is very much related to the theory of inverse problems, where the

The goal is precisely to recover information about a medium, you know, inside out of measurements

done on the boundary, right?

And this is the basis of many of the medical applications, like, you know, scanning tomographies,

PET, etc.

Right?

So we have all seen these images in which, for instance, one can get very clear pictures

of our brain or any other organ or bones inside out of signals taken from the exterior, right,

boundary.

So this is precisely one of the starting points of this relevant topic of inverse problems

for wave equations.

This is, of course, very important as well when you are considering the, you know, the

search and the management of, you know, natural resources like oil or water underneath, right?

So we are sending signals from the surface of Earth in order to detect what is inside.

And you see that the principle, I mean, just a small discussion before I get into the proof,

right?

The principle of this kind of inverse problems or inverse problems or solution techniques

is precisely that in case, right, this is more or less what we said yesterday, right?

So you are trying to analyze what is inside this medium omega to which you cannot penetrate,

right?

So you live here on the boundary.

So we are living here on the boundary of this domain outside, right?

And we wonder what is inside.

So is there water, petrol, granitic drug, whatever?

So the principle is that you say, okay, fine, in case, right, in case I send a signal, right,

in case I send some kind of pulse ray, right, it will penetrate the domain, right?

And in case the medium is homogeneous, this ray will, you know, exit the domain on the

other side, right?

So in case this set were empty somehow, right, you will send a signal here.

And then you will see something coming out over there, right?

Actually you could even, according to the travel time from here to here, you could also

measure what is the medium under consideration, right?

Because if you consider this wave equation here where the coefficients are the unit,

right, then you can always say, oh, yeah, then, you know, the coefficients of your equation

are one, the velocity of propagation is one.

But when you change the coefficient, you change, for instance, the density here of

the wave equation, then the velocity of propagation changes.

And then this travel time, right, this travel time that is something that you can record,

right, travel time, will give you already information about the medium that you are

trying to discover, disclose, right?

But it could also happen that this medium is not homogeneous, right?