## Agregation 2013 – Analysis – Part 4

**Part IV: To be or not to be with separable dual**

1. Consider a segment () of and a closed linear subspace of such that each function in is of class on . For such that we denote where .

(a) Prove that .

(b) Prove that for every we have

(c) Deduce that there exists which verifies for each and

(d) Let be a finite sequence of points in such that (for ) and . Prove that

2. Let be a closed linear subspace of such that every function in is of class on . Prove that has finite dimension.

## Characterizations of Sobolev Spaces

I will present here a few useful characterizations of Sobolev spaces. The source is *Functional Analysis, Sobolev Spaces and Partial Differential Equations* by Haim Brezis. Consider with . We denote by the conjugate of , i.e. . Then the following properties are equivalent:

- (i) ;
- (ii) there exists a constant such that
;

- (iii) there exists a constant such that for all and all with we have
(note that makes sense for and . Furthermore, we can take in (ii) and (iii). If we have

.

## Weak formulation for Laplace Equation with Robin boundary conditions

Consider an open set with Lipschitz boundary and consider on the following problem

where is a constant. This is the Laplace equation with Robin boundary conditions. I will prove that the problem is well posed and for each there exists a solution .

## Lax Milgram application

Let and . Consider the bilinear form

1. Check that is a continuous symmetric bilinear form and that implies .

2. Prove that is coercive.

3. Deduce that for every there exists a unique satisfying

What is the corresponding minimization problem?

4. Show that the solution of belongs to (and in particular ). Determine the equation and the boundary conditions satisfied by .

5. Assume that , and let be the solution of . Prove that belongs to for every . Show that if and only if .

6. Determine explicitly the solution of when is a constant.

7. Set , where is the solution of and . Check that is a self-adjoint compact operator from into itself.

8. Study the eigenvalues of .

*H. Brezis, Functional Analysis*

## Uniqueness and Error estimates via Kinetic Entropy Defect Measure

Here are a few thoughts from my preparation for the exam of Kinetic Equations at Universite de Savoie, France. The teachers of the course were Christian Bourdarias and Stephane Gerbi. I had to study an article of Benoit Perthame entitled *Uniqueness and Error estimates in First Order Quasilinear Conservation Laws via the Kinetic Entropy Defect Measure*.

This was a very nice article to study, since it used many things like distribution theory, measures and regularization. It showed the power of these tools, and motivated me to learn more about them.

As the title of the article says, a relatively new proof of the uniqueness of the solution for a scalar conservation law coupled with some entropy inequalities is given. The only known proof at the time the article was published was due to Kruzkov and was more intricate and difficult to understand than the one provided in the article. The estimates on the entropy defect measure, which will be introduced can yield some error term approximation for approximate equation, which in particular imply unicity at once.

Here are my detailed notes on the article. They are handwritten, but I think they are readable. Perthame-Uniqueness and Error Estimates