Rearrangement minimization problems with indefinite external forces

This paper discusses a rearrangement minimization problem related to a boundary value problem where the differential operator is the p-Laplacian and the external force is indefinite. By applying the rearrangement theory established by G. R. Burton, we show that the minimization problem has a unique solution in the weak closure of the admissible set in an appropriate function space. What makes this paper different from many others in the literature is that the admissible set here is allowed to be indefinite (takes on both positive and negative values). We show that in this case the optimal solution will be a weak limit point of the admissible set and cannot be a rearrangement element itself. Optimality conditions are derived based on the mean value of the generator. In case the generator is a three valued function, we shall show that the optimality conditions lead to free boundary value problems that are of interest independently. We also discuss a minimization problem where the admissible set is not seemingly related to rearrangement classes but we shall prove otherwise, and apply the earlier results of the current paper to draw interesting conclusions regarding the optimal solution. We have also discussed the case where the domain of interest is radial, and shown that the optimal solutions are radial as well.