Tamtam Proceedings - lamsin

462 Jaoua et al.1. IntroductionAmong data extension issues in elliptic inverse problems there arises the task of recoveringeither Dirichlet (or Neumann) boundary data (temperature field, electric potential,...), a Robin type exchange coefficient, or a crack located on some interfaces of the structure,from overdetermined measurements on the outer boundary. In the case of a tube oron pipe, the problem should reduce to a 2D one on an annulus.This can also be related to the inverse electroencephalography problem in spherical 3Ddomains, which give simple models of the human head, assumed to be a ball made of(at least) three concentric spherical homogeneous layers (brain, skull, and scalp) [4][2]. Overdetermined electrical measurements (potential, current flux) are available onthe scalp (external boundary), from which one wants to recover some current sources(conductivity defaults) located in the brain (inner layer), the potential being harmonicelsewhere in the domain. Taking planar cross-sections of the ball allows one to expressthe problem in a family of discs, where the above 2D data extension issue may arise as apreliminary step before recovering the singularities.These issues, and some others, usually involve solving a Cauchy problem for the Laplaceoperator on an annulus, from available data on the other boundary. This problem is knownto be ill-posed since the work of Hadamard, its most critical feature being the lack of continuityof the solution with respect to the data – whenever it exists. (This is the case forcompatible data, which means that the overdetermined data is indeed the trace and normalderivative of the solution of a single harmonic functional.) Therefore great care isrequired when solving such a problem. There are several general algorithms for solvingthese problems or dealing with applications close to those intended in this work [3]; howeverall these algorithms solve effectively the Cauchy problem (i.e., we have the solutionin the entire domain). They are based on multiple resolution of the backward problemand are therefore time consuming. Our approach is cheap and based on harmonic approximation[5] consisting in building from a flux imposed and the temperature measured onthe part of the outside border of the annulus, a complex function defined on the surface ofmeasure and find an extension of it defined on G.2. Approximation in Hardy classesLet D be the unit disc and G be the annulus G = D \ sD for some fixed s with0 < s < 1. The Hardy spaces H p (G), 1 ≤ p < ∞, on a circular domain G weredefined by Rudin [1] in terms of analytic functions f such that |f(z)| p has a harmonicmajorant on G, that is, a real harmonic function u(z) such that |f(z)| p ≤ u(z) on G. Itis also possible to define the Hardy spaces H p (∂G) for 1 ≤ p < ∞ as the closure inL p (∂G) of the set R G of rational functions whose poles lie in the complement of G. Thespaces H p (G) and H p (∂G) are then isomorphic in a natural way, and so we identify theTAMTAM –Tunis– 2005