On few geometrical inverse problems by way of elliptical conduction equations: theoretical and numerical study

We consider in this work a class of identification problems defined via a physical phenomenon described by elliptic partial differential equation (thermal diffusion..). Provided dual quantities being 'measured' on the boundary of a body at equilibrium (temperature and normal outwards flux... ), the question is to derive information on the internal structure of the body. We focus here on the problems of detection of cracks, voids, unknown boundaries... From a mathematical viewpoint these problems pertain to inverse geometrical problems. We prove uniqueness results by one over specified data in the case of an unknown boundary, a crack which has an endpoint on the boundary (2D) and a planar crack (3D). For numerical treatments, stability results are proved: we prove that the mapping which associates the geometry to the data is locally Lipschitzian. The numerical part of this work is devoted to line segment crack determination. Based on a semi-iterative method, an identification process in two steps is performed: in the first step the crack plane is directly located by an inversion formula; in the second step, we complete iteratively the crack identification by minimizing a functional initially introduced by Kohn and Vogelius for parameters identification. (author)