Rearrangement minimization problems corresponding to $p$-Laplacian equation

In this paper a rearrangement minimization problem corresponding to solutions of the $p$-Laplacian equation is considered. The solution of the minimization problem determines the optimal way of exerting external forces on a membrane in order to have a minimum displacement. Geometrical and topological properties of the optimizer is derived and the analytical solution of the problem is obtained for circular and annular membranes. Then, we find nearly optimal solutions which are shown to be good approximations to the minimizer for specific ranges of the parameter values in the optimization problem. A robust and efficient numerical algorithm is developed based upon rearrangement techniques to derive the solution of the minimization problem for domains with different geometries in $\mathbb{R}^2$ and $\mathbb{R}^3$.