Section: New Results
New results in the theory of factorization of boundary value problems
Participants : Jacques Henry, Maria Orey.
We are pursuing the development of the theory of factorization of boundary value problems as described in 3.1 . One of the main mathematical difficulties of this theory from the mathematical standpoint is that the full justification of the computation needs the study of a Riccati equation for unbounded operators. In previous papers it has been done by a Galerkin method or within the Hilbert-Schmidt framework. Both method are lengthy and technical. The Yosida regularization has been used successfully in the case of the Laplacian in a cylinder but the generalization to other operators and domains seems difficult. A new method has been investigated: a regularization of the original problem using the square of the operator tangential to the moving surface leads to a problem that can be directly interpreted as the optimality system for the optimal control of a parabolic equation. The justification of the Riccati equation for such a problem is now clearly established. This is also the case in non cylindrical domains from a paper by J. P. Zolesio. This work has been presented at the IFIP TC-7 conference in Buenos Aires.
As explained in 3.1 boundary value problems factorisation can be viewed as an infinite dimensional extension of the block Gauss factorisation for matrices. The block are related to the variable section. A natural question that arises then is what is the extension of the full Gauss factorisation. A preliminary investigation of this question on the Laplacian in a rectangle leads to a solution that needs the factorisation of the operator solution of the Riccati equation. This is also related to the spatial factorisation of parabolic problems.
The study of the QR factorization of boundary value problems, which is the subject of Maria Orey's thesis is pursued. A chapter of a book to be published in 2010 by Intech has been written jointly by M. Orey, B. Louro and J. Henry. Improved derivation and justification of the calculation of the factorisation of the normal equation of the problem set in the least squares sense is presented there.