Detecting nonlinear corrosion by electrostatic measurements

where Γ1 and Γ2 are two open, disjoint portions of ∂Ω and ΓD = ∂Ω \ (Γ1∪Γ2) and f is a Lipschitz function such that f(0) = 0. Such a problem with a specific choice of the profile of f , has been introduced and discussed in recent years by M.Vogelius and others for the modeling of the electrochemical phenomenon of surface corrosion in metals [7],[13],[20]. In the above boundary value problem, Ω represents the metal specimen, Γ1 represents the corroded part of the boundary, which is not accessible to direct inspection, Γ2 is the portion of the boundary which is accessible to measurements and ΓD is the remaining part of ∂Ω which is assumed to be grounded. The inverse problem thus consists in the determination of f when one pair of Cauchy data u|Γ2 , ∂u ∂ν |Γ2 is known for one non-trivial solution u to (1.1). It must be noted that the models of nonlinearity which have been discussed in [13],[20] are of the form