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In this work, we consider the inverse problem of identifying a Robin coefficient in a nonlinear elliptic equation with mixed boundary conditions. We firstly reformulate the inverse problem as a regularized optimal control one, where the functional cost is of typeL1-L2L^{1}-L^{2}; then we prove the existence and uniqueness of a minimizer to the resulting optimization problem in a suitable functional space. Finally, we provide a primal-dual algorithm to solve the variational problem and give some numerical results that prove the accuracy of the proposed method in the identification of the Robin coefficient.

A primal-dual approach for the Robin inverse problem in a nonlinear elliptic equation: The case of the 𝐿 1 − 𝐿 2 cost functional