Optimization of the energy integral in two classes of rearrangements

This paper concerns the maximization and the minimization of the energy integral associated to a second order elliptic problem in classes of rearrangements with respect to either the Lebesgue measure or to a measure related to the structure of the equation. We find results of existence, uniqueness and representation of the maximizers and the minimizers. Precise characterizations of the optimizers are found in case the domain is a ball. Finally, the effect of special geometrical transformations concerning these problems is discussed.