Okay.

Okay, first of all I want to thank

Professor, and all the stuff in the, in the chair of applied analysis here in the lung and I'm very glad to be here to stay here in spirit.

And I thank very much.

And all the stuff for the warm hospitality here. So it's a, it's a pleasure to be here. And today.

I will talk about this, the title is shape of position and as a very medical inequalities. I will.

In the first part of the talk I will give some basic and

basic insight on what is a shape of the position problem and some very classical results.

And at the end I will give some detail on some recent work. I did.

Okay, so,

just a moment.

I cannot change. Okay.

So what is a ship optimization problem so the aim is to minimize or maximize the functional, which depend on on some shape, omega

and this functional.

So is defined on sets that leaves in some class of admissible shapes so on on you you can on omega you can give some constraint.

Like for example, complexity open sets measure measurable sets and so on, and or simply connected and some constraint on, for example, the volume or the perimeter so you can fix several parameters.

And so the aim is to minimize or maximize this functional.

So in many applications, the functional F depends on omega via the state function so function you omega, which arises as a solution of some partial differential equation, given in omega.

And so, to basic words in this, in this, in this area. The first one is isometric inequalities which is obviously related the first says the well known is a very magic inequality, but nowadays is it identifies

problems that regard the geometric quantities the diameter so different notion of parameters or physical quantities as eigenvalues so boundary value problems so torsional agility and so on.

And another keyword is the word symmetry station. And what is this an additional procedure is a meditation procedure consists roughly speaking in transformer mathematical object like sector of function in one in another one more symmetric

that actually preserves some property of the original object. So,

as I, as I told the first classical example about symmetry station and shape optimization is the problem of the classical is a very magic inequality so if you have a set omega.

And you consider the Rn and, and you consider the ball that I denote with omega sharp with the same volume, then omega.

Then, the this ball as smallest, the smallest perimeter. So the ball omega sharp is the solution of the of the shape optimization problem minimize the perimeter among all the sets of fixed volume.

And

the whole the ball is the minimize.

And, okay, this is a, this is a meditation procedure because if you consider the procedure that from a set omega goes in the ball omega sharp with the same volume, we see that this procedure minimize the perimeter.

And another symmetry station procedures about functions. So if you consider a function you.

And you consider them the function, you sharp, which is the so called the spots symmetries function of you, and I was constructed you sharp you sharp is the function, which

whose level sets are concentric bolts with the measure equal to the measure of the corresponding level set of you. So, if you consider the super level set you greater than T.

Then the corresponding level set of the function you sharp is a ball with the same volume then you greater than T centered at your aging. So, this is a one dimensional okay two dimensional graph but you can imagine in all dimension.

And so, for every level set you you do this this construction.

And so this procedure is named as a box rearrangement or spherical decreasing the arrangement, it is in general, the written as you sharp.

You sharp is this function which is rather symmetric decreasing and the measure the level sets is the same as the same measure of the level sets of you and.

And are concentric bolts.

So you sharp is defined on the ball omega sharp, which has the same volume then omega.

Obviously we are considering the set omega with finite volume.

So what what what what this procedure.

What we gain this procedure. So the first thing is that we have by Cavalieri principle, we have that LP norm are preserved so function you and you sharp have the same LP norm for any.

And the second fundamental tool is the fact that the energy decrease so if you have a function you with compact support so function in you.

H01 omega H1 in all our end in with compact support so the the energy decreases so the gradient of you sharp square is smaller is less than the gradient of view square in our end.

And this is the, the polio.

Zero inequality.

And okay I have written here with power two but actually this holds for any for any P.

Okay, so as I mentioned, I want to recall two classical problems. So the first one is

regards the solution of

the partial differential equation minus delta u equal to F in omega with the boundary condition, you will see on the omega.

So,

problem that we can ask is the following. So, among all problems, P.