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 $BV(\Omega)\cap L^\infty(\Omega)$. 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 $\varepsilon$-$p$-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 $\varepsilon$-$p$-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 $(\varepsilon,k)$-level as the parameters tend to zero and infinity, respectively.