Approximation of an Optimal BV -Control Problem in the Coefficient for the p(x)-Laplace Equation

Abstract

We study a Dirichlet optimal control problem for a quasilinear monotone elliptic equation with the so-called weighted p(x)-Laplace operator. The coefficient of the p(x)-Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the admissible controls. In order to handle the inherent degeneracy of the p(x)-Laplacian, we use a special two-parametric regularization scheme. We derive existence and uniqueness of variational V -solutions to the underlying boundary value problem and the corresponding optimal control problem. Further we discuss the asymptotic behaviour of the solutions to regularized problems on each (ε, k)-level as the parameters tend to zero and infinity, respectively. The characteristic feature of the considered OCP is the fact that the exponent p(x) is assumed to be Lebesgue-measurable, and we do not impose any additional assumptions on p(x) like to be a Lipschitz function or satisfy the so-called log-H‥older continuity condition

Keywords

• nonlinear dirichlet problem
• variable exponent
• p(x)-laplace operator
• optimal control
• control in coefficients