Constrained and unconstrained rearrangement minimization problems related to the p-Laplace operator

In this paper we consider an unconstrained and a constrained minimization problem related to the boundary value problem

$$ - {Delta _p}u = f{text{ in }}D,{text{ }}u = 0{text{ on }}partial D$$

. In the unconstrained problem we minimize an energy functional relative to a rearrangement class, and prove existence of a unique solution. We also consider the case when D is a planar disk and show that the minimizer is radial and increasing. In the constrained problem we minimize the energy functional relative to the intersection of a rearrangement class with an affine subspace of codimension one in an appropriate function space. We briefly discuss our motivation for studying the constrained minimization problem.