Recovering unknown terms in a nonlinear boundary condition for Laplace's equation

We consider a thin metallic plate whose top side is inaccessible and in contact with a corroding fluid. Heat exchange between metal and fluid follows linear Newton's cooling law as long as the inaccessible side is not damaged. We assume that the effects of corrosion are modeled by means of a nonlinear perturbation in the exchange law. On the other hand, we are able to heat the conductor and take temperature maps of the accessible side. Our goal is to recover the nonlinear perturbation of the exchange law on the top side from thermal data collected on the opposite one (thermal imaging). In this paper we use a stationary model, i.e., the temperature inside the plate is assumed to fulfill Laplace's equation. Hence, our problem is stated as an inverse ill-posed problem for Laplace's equation with nonlinear boundary conditions. We study identifiability and local Lipschitz stability. In particular, we prove that the nonlinear term is identified by one Cauchy data set. Moreover, we produce approximated solutions by means of an optimizational method.