On the corrector term in the homogenization of the nonlinear Poisson-Robin problem giving rise to a strange term: Application to an optimal control problem
Jesús Ildefonso Díaz Díaz, Alexander Podolskiy, Т. А. Шапошникова · Journal of Mathematical Analysis and Applications · 2024
We consider the homogenization process corresponding to some heterogeneous problems that are given by the Poisson equation with a nonlinear Robin-type boundary conditions on the interior boundary of some small perforations (or the boundary of some small particles) in the so-called critical case, giving rise to the appearance of a strange term in the limit semilinear equation. We prove the strong convergence, in the corresponding Sobolev space , of the solutions with a suitable corrector term. In contrast with other previous results in the literature, we do not assume any additional regularity on the solution of the limit equation: we prove that when the spatial dimension is n = 3 or n = 2 , then the inherent H 2 regularity is enough to get such a strong convergence. As an application we consider an optimal control problem in which the cost functional involves the gradient of the state solutions, being independent of the nonlinear term arising in the Robin boundary conditions. By working with the corresponding adjoint problem, we show that the limit of the optimal controls is given as a suitable optimal control associated with the new cost functional in which the strange term and other related terms arise in some unexpected way.