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NEW RESULTS ON THE NON-LINEARITY AND THE SENSITIVITY OF THE ESTIMATION OF THE DIFFUSION COEFFICIENT IN A 2D ELLIPTIC EQUATION

We consider the problem of determining the diffusion coefficient a x in a 2D elliptic equation from a distributed measurement z in H 1 of the solution u of the equation. For a problem with a simple geometry, we give conditions under which the first derivative of the b 1 a u mapping is coercive. Then we show that its non linearity in a direction d increases, and its sensitivity decreases, when the ratio ∇ d b L2 d b L2 increases. This corroborates observations on scale, sensitivity and non linearity made in (Chavent and Liu, 89) (Grimstad and Mannseth, 99).