Good morning or good afternoon or good evening, depending on where you are, right?

So nowadays these events are global.

We are delighted to welcome Alessio Figali to our Maud Colloquium, Maud Lectures.

So our university, Frederic Alexander University in Erlangen-Nuremberg, is not only running activity in mathematics,

but actually we have a very strong agenda in engineering and also with applications to medicine, the medical valley.

And then the university decided to gather a number of us on this research center, Mathematics of Data,

because we are, of course, as many other institutions also,

involved in the field of data science, artificial intelligence and machine learning,

not only from the math perspective, but also from the applications.

And one of the activities we try to promote is to run once every, say, quarter or so.

So we are one of these Maud Lectures, trying to bring the best people in our field.

Today we are very honored that Alessio accepted to deliver this talk, this time online.

I think he's someone that nobody needs many details.

You can have a look in the Wikipedia.

Alessio was born in 84 in Rome, so he's still a very, very young mathematician.

Despite of this, his records and trajectory are very impressive.

He was a student with Luigi Ambrosio and Cedric Vilani.

He has been also a close collaborator of Luis Cafarelli, recent Avel Prize winner.

And he got himself the Fils Medal a few years ago, I think in 2018.

So it's already five years ago that he got it.

For the last few years he has been a professor in ETH Zurich.

And among the different topics he has developed, probably those that are better known for our community are

optimal transport, Mons-Rampert equations and free boundary problems among other fields.

And today he's precisely lecturing on free boundary problems that, as you know, is one of these classical topics

in analysis of PDEs that is certainly one of the most challenging and on which still,

despite of the great progress and contributions in particular by Alessio, there is still so much to be done.

So this is why we believe that the topic chosen by Alessio for this lecture is particularly appropriate.

So Alessio, thank you again. And we are ready to listen to you. Thank you.

Thank you very much, Eric. It's really, really a pleasure for me to deliver this lecture.

So, yeah, today I would like to talk about the obstacle problem, which is a very classical free boundary problem.

I mean, many of you already know it, but I prefer to start already from the beginning and really have, I mean, you don't need any knowledge of it.

I will guide you through this problem and guide you through the free boundary regularity. That's the goal of this lecture.

So what is the obstacle problem? There are many ways to introduce it. It models a lot of different phenomena.

I will take away an approach which is kind of convenient for me to get immediately to the back and very quickly to the equations that we have to solve.

And it starts as follows. Let's say that I have a wire. OK, so this is a metal wire.

So this is a wire that I assume I can, it's a graph of some function over some domain.

So let's say that I have some domain here downstairs that I denote by omega.

And then let's say I have some function. This wire can be seen as the graph of some function.

So I have some function f defined on the boundary of omega with values in, let's say, 0 plus infinity.

So I assume the function to be positive. And my wire is the graph of f of this function.

OK, so what do I do? My metal wire is here in the space, lying in my space. And I'm going to attach an elastic membrane to it.

So I will have now an elastic membrane that is going to be attached to my wire.

So this is my elastic membrane.

And so my elastic membrane is what? It's again a graph. So this is a graph of some function u.

Well, now u is a function from omega into r.

And the fact that u is attached to the wire means that u restricted to the boundary of omega is equal to f. So this is my boundary condition.

Now, my goal is to find the equilibrium configuration of an elastic membrane attached to a wire.

So let me start my steps. So the first step is, if you want, just basic linear elasticity.

So step one, point. What I'm going to do is going to minimize an energy.

And then I'm going to put some will consider all functions that are equal to f on the boundary.