Maximization problems are solved for Voiculescu's free entropy of probability measures supported in R, R+, and [ - 1,1], respectively, under constraint of the pth moment for any p > 0 and implications of these results for multivariate free entropy are discussed in the setting of noncommutative random variables. Similar extremum problems are treated for probability measures on C and T under certain constraints. The elliptic law and a distribution found earlier in quantum physics are encountered. These results are in the setting of potential theory and can be viewed independently from Voiculescu's work. The machinery of weighted potentials is exploited.
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