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We consider a system y t (t, x) = −Ay(t, x) + g(t, x)x ∈ with a suitable boundary condition, where ⊂ R n is a bounded domain, −A is a uniformly elliptic operator of the second order whose coefficients are suitably regular for (t, x), θ ∈]0, T [ is fixed, and a function g(t, x) satisfiesOur inverse problems are determinations of g using overdetermining data y |]0,T [×ω or {y |]0,T [× 0 , ∇y |]0,T [× 0 }, where ω ⊂ and 0 ⊂ ∂ . Our main result is the Lipschitz stability in these inverse problems. We also consider the determination of f = f (x), x ∈ in the case of g(t, x) = f (x)R(t, x) with given R satisfying R(θ, •) > 0 on . Finally, we discuss an upper estimation of our overdetermining data by means of f .

Small-angle scattering (SAS) experiments on SI GaAs crystals reveal that: (i) in GaAs there exist two regions of altered optical properties surrounding dislocations, the inner region having a radius of 5 p m and the outer region a radius 30-40,um; (ii) these regions are formed by As-related defects such as V G~, As, and AsG.; the inhomogeneities in question contain centres with an ionisation energy 0.1 -0.2 eV and quite shallow centres a s well. These inhomogeneities give rise to a considerable conductivity dependence on frequency.

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We prove for a two-dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary uniquely determines the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can uniquely determine the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.