Maximax and minimax rearrangement optimization problems

In this paper we present a general theory concerning two rearrangement optimization problems; one of maximization and the other of minimization type. The structure of the cost functional allows to formulate the two problems as maximax and minimax optimization problems. The latter proves to be far more interesting than the former. As an application of the theory we investigate a shape optimization problem which has already been addressed by other authors; however, here we prove our method is more efficient, and has the advantage that it captures more features of the optimal solutions than those obtained by others. The paper ends with a special case of the minimax problem, where we are able to obtain a minimum size estimate related to the optimal solution.