Minimization of the Compliance under a Nonlocal p-Laplacian Constraint

Abstract

This work is an extension of the paper by Cea and Malanowski to the nonlocal and nonlinear framework. The addressed topic is the study of an optimal control problem driven by a nonlocal p-Laplacian equation that includes a coefficient playing the role of control in the optimization problem. The cost functional is the compliance, and the constraint on the states are of the Dirichlet homogeneous type. The goal of the present work is a numerical scheme for the nonlocal optimal control problem and its use to approximate solutions in the local setting. The main contributions of the paper are a maximum principle and a uniqueness result. These findings and the monotonicity properties of the p-Laplacian operator have been crucial to building an effective numerical scheme, which, at the same time, has provided the existence of optimal designs. Several numerical simulations complete the work.