Approximation of optimal control problems in the coefficient for the p-Laplace equation. I. Convergence result

We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the so-called weighted p-Laplace problem. The coefficient of the p-Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the control such that there is a strictly positive lower and some upper bound. In order to handle the inherent degeneracy of the p-Laplacian, we use a regularization, sometimes referred to as the ε-p-Laplacian. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted ε-p- Laplacian, where we approximate the nonlinearity by a bounded monotone function, parametrized by k. Further, we discuss the asymptotic behavior of the solutions to the regularized problem on each (ε, k)-level as the parameters tend to zero and infinity, respectively.This author’s research was supported by the DFG-EC315 “Engineering of Advanced Materials” and by the Spanish Ministerio de Economía y Competitividad under projects MTM2011-22711 and MTM2014-57531-P