Abstract. 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-pLaplacian, 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. 1. Introduction. Control in the coefficients of elliptic problems has a long history of its own, starting with the work of Murat [10,11] and Tartar [14]. The constrained optimal control problem (OCP) in the coefficients of the leading order differential expressions was first discussed in detail by Casas [2] in the case of the classical Laplace equation, where the scalar coefficient u in the div(u∇·) formulation was taken as control satisfying box constraints with a strictly positive lower and some upper bound together with a slope constraint. The problem of existence and uniqueness of the underlying boundary value problem and the corresponding OCP was treated, and an optimality system has been derived and analyzed. Analogous results for the case of general quasi-linear elliptic equations of the type div (a(u, ∇·)) remained open. In this article we treat the case of the weighted p-Laplacian, where a(u, ∇y) = u|∇y| p−2 ∇y. The corresponding quasi-linear differential operator, − div(u|∇y| p−2 ∇y), in principle, has degeneracies as ∇y tends to zero and also if u approaches zero. Moreover, when the term u|∇y| p−2 is regarded as the coefficient of the Laplace operator, we also have the case of unbounded coefficients. In order to avoid degeneracy with respect to the control u, we assume that u is bounded away from zero. For the precise statements, see the next section. We leave the case of potentially degenerating controls to a future contribution. Instead, in this article, we focus on the degeneracies related to the nonlinearity. A number of regularizations have been suggested in the literature.
