A nonlocal free boundary problem with Wasserstein distance

We study the probability measures $\rho\in \mathcal M(\mathbb R^2)$ minimizing the functional $$J[\rho]=\iint \log\frac1{|x-y|}d\rho(x)d\rho(y)+d^2(\rho, \rho_0),$$ where $\rho_0$ is a given probability measure and $d(\rho, \rho_0)$ is the 2-Wasserstein distance of $\rho$ and $\rho_0$. % We prove the existence of minimizers $\rho$ and show that the potential $U^\rho=-\log|x|\ast \rho$ 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-Amp\'ere equation, which after linearization leads to the equation $\rho_t={\hbox{div}}(\rho\nabla U^\rho)$. The methods we develop use Fourier transform techniques. They work equally well in high dimensions $n\ge2$ for the energy $$J[\rho]=\iint |x-y|^{2-n}d\rho(x)d\rho(y)+d^2(\rho, \rho_0).$$