## Section: New Results

### New results in the theory of factorization of boundary value problems

Participants : Jacques Henry, Fadhel Jday, Maria Orey.

We are pursuing the development of the theory of factorization of boundary value problems as described in 3.1 . Maria Orey who suspended her PhD thesis for a while due to health reason, has resumed her work on extending the method of factorization to the analogous in infinite dimension of the $QR$ algorithm for matrices. This passes through the factorization of the normal equation for the least squares problem. This problem is solved and this allows a clear definition of the $Q$ and $R$ operators. She will defend her thesis in 2012.

F. Jday has obtained also a clear formulation for the factorization of the Stokes equation.

A progress has been made in the attempt to extend the factorization method to parabolic evolution equation. It appears that it is not the evolution problem that can be factorized with respact to space but the evolution operator $S\left(t\right)$ that transfers the solution from time 0 to $t$. The corresponding Riccati equation has been obtained but a full mathematical justification remains to be done.