Constrained energy problems with applications to orthogonal polynomials of a discrete variable

Given a positive measure sigma with parallel to sigma parallel to > 1, we write mu is an element of M-sigma if mu is a probability measure and sigma - mu is a positive measure. Under some general assumptions on the constraining measure sigma and a weight function w, we prove existence and uniqueness of a measure lambda(w)(mu) that minimizes the weighted logarithmic energy over the class M-sigma. We also obtain a characterization theorem, a saturation result and a balayage representation for the measure lambda(w)(sigma). As applications of our results, we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials.