Carl Friedrich Gauss investigated in the variational problem of minimizing the Newtonian energy evaluated in the presence of an external field, nowadays called the Gauss functional, over nonnegative charges on the boundary surface of a given domain. For this problem, later on the sign condition was given upin connection with boundary integral equation methods where distributional boundary charges had been introduced for solving boundary value problems. A different generalization of the original Gauss variational problem, maintaining the sign restriction but employing Borel or Radon measures as charges and replacing the Newtonian kernel by the much more general Riesz kernel has independently grown into an eminent branch of modernpotential theory. In this project, we analyize the related minimization problem and develop efficient techniques to numerically compute the solution.