Optimal shapes in force fields

We study a variational problem involving material shapes, modeled as Radon measures on a given ambient space, in a force vector field. We prove some properties of absolutely continuity of the optimal shapes in dependence on the distributions of the zeros of the force field. More regularity properties of the optimal shapes, as well as simple geometrical characterizations, are established in the case of central force fields, in dependence on the regularity of the field and on the choice of the ambient space.