So welcome everyone again to this week, the online seminar.

Today we have Professor Giuseppe Batuzzo from the University of Pisa in Italy.

And he's speaking about the relation between principal immune value and torsional rigidity.

Please. Yes. Thank you. Just a moment because I'm low in battery. Okay. Now it's okay.

So I'm very happy to speak in this chair of applied analysis seminar.

And this is a good opportunity for me to show this research about the relation between the

principal immune value and torsional rigidity. So let me say that this is a joint research

made with the Mikkel Vandenberg from Bristol and Aldo Pratelli who was in Erlangen first now is in

my university in Pisa. And I will show some numerical experiments made by a student of mine,

my Ginevra Bionde. She was a master student at the University of Pisa. Now she finished.

Okay. The goal is to show the two quantities that are very, very common when you study elliptic

equation. Well, elliptic equation means elliptic operators, but I will take always the Laplace

operator, which is a very good prototype. And also for simplicity, I will take Dirichlet boundary

conditions. This is by strong simplicity because Dirichlet boundary conditions are much, much

simpler than the other ones. Well, other operators is not a problem. If you want to replace a Laplace

operator by another one, well, this is not very, very complicated. On the contrary, if you replace

Dirichlet boundary conditions by Neumann on Roubaix conditions, you will have several extra

difficulties. And this is essentially due to the extension problem because

Sobolev spaces with zero boundary conditions can be immediately extended to all the space. You put

zero outside and you obtain a Sobolev function defined on all the space. But this is not possible

when you deal with Neumann or Roubaix boundary conditions. So this is the main

difficulty. So from now on, everything will be with Dirichlet boundary condition. But also in

this case, you will see there are some very interesting open problems. So just to start with,

to introduce these two quantities, let us start by making two experiments, two measurements.

So omega is the body. The model is the heat source diffusion. So omega is the body. In omega,

we take the uniform heat source, which means the right hand side F equal to one,

fix an initial condition to start with zero boundary condition at the boundary, Dirichlet

boundary condition, and let the heat flow. So this is the first experiment.

Let the heat flow. Wait for a long time. And finally measure the average temperature of the body.

This is a number. So you relate this number, this average temperature, to the omega. So for every

omega, you have this number, which is the temperature, the average temperature of the body omega.

Now the second number is obtained in a similar way. Now omega is always the body.

This time you have no heat source, so F equal zero. Clearly, well, it is evident that if you

start from an initial temperature U zero, and you let the heat flow, since there is no heat sources,

and since you consider the Dirichlet boundary condition, the temperature will go to zero.

Clearly, the temperature will go to zero. But you are interested to measure how fast

this temperature goes to zero. So you measure this time the decay rate to zero.

So the first measurement, you measure the average temperature. In the second experiment,

you measure the decay rate to zero. And the question is to study how these two numbers

are related to each other. So this is the problem I want to

deal with. So the first quantity is usually called the torsional rigidity, because people like more

to speak about the mechanical problems. But you know that a plus operator enters in many

simulations, so I prefer to speak about the heat.

But exactly the same, you can rephrase the problem in a mechanical way.

So torsional rigidity, what torsional rigidity is, is the integral of the solution U of the PDE minus

the Laplacian U equals one. And this is the heat equation when you wait for a long time.

When you wait for a long time, the heat equation becomes stationary. So this is minus Laplacian U

equals one in omega and zero boundary conditions. So this has a unique solution. Take the unique

solution, take the integral, and this is what we call torsional rigidity. It's very simple.

So in the thermal diffusion, this T of omega over the measure of omega is the average temperature,