Okay, so welcome everyone to the CA09 seminar. Today we have Professor Jose Rodriguez from

the University of Lisbon and he will be talking about unconvex type constraint problems of

false order and some applications. We are looking forward to your talk, Professor Rodriguez.

Okay, thank you very much for this opportunity to revisit an old topic. So I will try to do a survey

of some of my work, but basically classical problems with convex constraints for first order

linear and non-linear operators. So they are likely to transport

equation and conservation law. And basically you see there are three items in my proposal talk.

In fact, this topic has been around for several decades. And as you may know, my favorite problem

is the obstacle problem. And you'll see that it's still for first order PDEs is a very interesting

challenge. Marius, if there is any problem with internet, let me know. Okay.

Is it okay? Yes, it's okay. I will let you know in case of any problems. It's fine.

Right. Okay. So I will divide my talk basically in three parts. The first one is obstacle problems

for transport operators. Then I will make a different type of convex constraint problem

for the gradient. So you constrain the gradient, not the solution itself. And finally, I will

try just to start with the very challenging and still partially open problem, which is the obstacle

mass constrain problem for scholar conservation law. And you will see what will be the difficulty

of understanding first the easiest cases in the transport operators or linear operators.

And then to see how things can evolve in these kinds of problems that have been treated by several

authors, but up to now, not yet, in my opinion, sufficiently developed. So they are very much

challenge problems also in not only theoretical, but also numerically. And for instance, the control

problems with this kind of setting is mainly open as far as I'm aware. Okay. You see, there are some

motivations, the polyphasic flow in porous media, populations model. I will not speak today. The

transports and pile problems is a very, very nice problem that we'll speak in a moment. And also

another class of problems with the superconductivity where there is a constraint on the curl, but I

will not speak today on this. Okay. Let's go back to my one of my favorite problems,

the obstacle problem, which, as you know, is looking for a solution of some PDE

that should lie above an obstacle psi. So psi is a given function. And when you have a solution,

that usually you have a free bounder, this blue line in the interior. And of course,

the separating the region where U is equal to psi equal to the obstacle, where is in this case,

strictly above the obstacle. And the PDE can usually, I mean, initially the PDE was just plus

just Laplace equations for litig problems, but you can consider parabolic hyperbolic problems

and even no local problems and et cetera. So today I will concentrate myself in this kind of

transport operator, start with transport operator, which is the simplest case. Of course, it is in the

domain in Q in RN. This Q can be the cylindrical, if you want to distinguish the ends or N plus one

variable, depending what you choose. And in fact, in this case, it could be considered that

and the T special variable, but you see a first order operator linear ingredient and linear on the

lower order. And this is a classical transport operator. And just, I'm going to concentrate here

some contributions I did already century. And there are these two papers that are here concentrated

after the problem has been introduced in 73 by Ben Sussan-Lyonce and developed for weak solutions

by Muenck in the seventies. In fact, as you can see here, there is two regimes, the regime where

the solution is equal to the obstacle and the solution where it is certainly above and transport

equation should be satisfied. Being a first order problem, we need to divide the boundary into parts.

We suppose the boundary leaps sheets. So this is the set where the normal is not defined, but there

is the inflow region that maybe could be the red part or the blue part as method, just a representation.

And in the inflow region concerning the flux B, which is a given vector field,

we can impose a kind of Greek Leak-Cauchy problem and boundary data that can be imposed in what I

call here gamma minus. And with respect to this formulation of the problem, as I told, this was

introduced in the framework of deterministic optimal control problems by Ben Sussan-Lyonce

long, long time ago. And it was started to be studied by Mignoux-Puel in 76. And then

they not only solved it, I mean, they were studying also as an extension and single