We study the probability measures rho in mathcal M(mathbb R^{2}) minimizing the functional J[ρ]=∬log1|x-y|dρ(x)dρ(y)+d2(ρ,ρ0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} J[\\rho ]=\\iint \\log \\frac{1}{|x-y|}d\\rho (x)d\\rho (y)+d^2(\\rho , \\rho _0), \\end{aligned}$$\\end{document}where rho _0 is a given probability measure and d(rho , rho _0) is the 2-Wasserstein distance of rho and rho _0. J[rho ] appears in aggregation models when the movement of particles is advanced by the potential -log |x|*rho . We prove the existence of minimizers rho and show that the potential U^rho =-log |x|*rho solves a degenerate obstacle problem, the obstacle being the transport potential. Every minimizer rho is absolutely continuous with respect to the Lebesgue measure. The singular set of the free boundary of the obstacle problem is contained in a rectifiable set, and its Hausdorff dimension is < n-1.Moreover, U^rho solves a nonlocal Monge–Ampère equation, which after linearization leads to the equation rho _t={text {div}}(rho nabla U^rho ). The methods we develop use Fourier transform techniques. They work equally well in high dimensions nge 2 for the energy J[ρ]=∬|x-y|2-ndρ(x)dρ(y)+d2(ρ,ρ0).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} J[\\rho ]=\\iint |x-y|^{2-n}d\\rho (x)d\\rho (y)+d^2(\\rho , \\rho _0). \\end{aligned}$$\\end{document}