Yeah.

Thanks.

Hey, so good morning, everyone.

Thank you again for so many call for setting up a talk.

So today, we will be listening to non local face coexistence model.

Please, personally, call you have the floor looking forward to your talk.

Thanks a lot.

It's a great pleasure for me to be in Erlang and at least virtually.

So during this talk, I will speak mainly about the Allen-Kahn equation, which is a famous

equation to describe face coexistence models.

I will partially review the classical case of the Allen-Kahn equation, and I will discuss

also the non local version of it based on the long range particle interaction model.

So I will consider also a very general situation in which the medium can be anisotropic.

And the main target is to deal with one dimensional symmetry and rigidity results.

All these broad settings related to a classical conjecture by Angel DeGiorgi that we will

also discuss in some detail.

And possibly we will see some sketch of the proofs of the main theorems that I will present.

So to introduce the non local Allen-Kahn equation, just let me remind the definition of fractional

Laplacian, so the fractional Laplacian is this strange integral here.

So the non local Allen-Kahn equation is the one I wrote here in which the fractional

Laplacian is equal to a bistable nonlinearity.

And the idea of the model is that you want to understand the situation in which you have

a region, say in the Euclidean space, in which every point has some kind of spin parameter

that can be either 1 or minus 1 in principle.

And every point is able to switch from 1 to minus 1, but so this free switching has somehow

a cost because points nearby influence your state somehow.

So the system somehow tries to minimize the change between the state 1 and minus 1.

So this is a very classical idea that goes back to van der Waals.

Basically, the idea is that you model this problem by a double well potential, which

has minima, say at 1 and minus 1.

And you add the penalization terms, an additional term, which is a small penalization, which

somehow prevents the system to change too fast from 1 to minus 1.

This is also related to ferromagnetism, for instance, in which you have a spin and you

don't want your material to change magnetization in a crazy way.

But for the sake of this seminar, I will focus on this simple equation here, and I will also

present some generalization later on.

So I take the fractional application equal to u minus u cubed, and s for me will be a

parameter between 0 and 1.

So among all the possible solutions, there are some special solutions that I'm particularly

interested in, which are the minimal solutions.

So the solutions that are local minimizers for the energy.

And while in principle I cheated a little bit, because when I described the model, I

said that you have an energy which is based by superposition of a double well potential

with a small penalization term.

But when I wrote the equation, there is no small parameter here.

But the two models are basically related, because if you start without a penalization

term, you can look at the large scale behavior.

So rescale your solution.

If you have a solution u, rescale the solution, calling it u epsilon, which is just u of x

over epsilon, which is a blowdown of your solution.