We consider the problem of determining the Robin coefficient of some specimen material, by performing measurements on some part of the boundary. An identifiability result is proved for Robin coefficients which are continuous functions with some negative lower bound. Both local and monotone global Lipschitz stability results are established. Finally, a cost function turning the inverse problem into an optimization one is proposed for numerical purposes. This function, which may be viewed as an energetic least-squares one, has an easy-to-compute Gderivative, which encourages us to consider implementing the gradient algorithm in forthcoming numerical experiments.

We establish some global stability results together with logarithmic estimates in Sobolev norms for the inverse problem of recovering a Robin coefficient on part of the boundary of a smooth 2D domain from overdetermined measurements on the complementary part of a solution to the Laplace equation in the domain, using tools from analytic function theory.

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We investigate an optimization problem (OP) in a non-standard form: the cost functional F measures the L 1 distance between the solution u ϕ of the direct Robin problem and a function f ∈ L 1 (M). After proving positivity, monotonicity and control properties of the state u ϕ with respect to ϕ, we prove the existence of an optimal control ψ to the problem (OP) and establish Newton differentiability of the functional F. As an application to this optimization problem the inverse problem of determining a Robin parameter ϕ inv by measuring the data f on M is considered. In that case f is assumed to be the trace on M of u ϕ inv . In spite of the fact that we work with the L 1 -norm we prove differentiability of the cost functional F by using complex analysis techniques. The proof is strongly related to positivity and monotonicity of the derivative of the state with respect to ϕ. An identifiability result is also proved for the set of admissible parameters ad consisting of positive functions in L ∞ .