A nonlocal free boundary problem with Wasserstein distance

Abstract

We study the probability measures $\rho \in \mathcal{M}({\mathbb{R}}^2)$ minimizing the functional \[ J[ρ]=\iint \log\frac1{|x-y|}dρ(x)dρ(y)+d^2(ρ, ρ_0), \] 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-Ampére equation, which after linearization leads to the equation ${\rho }_t=div(\rho \nabla U^{\rho })$. The methods we develop use Fourier transform techniques. They work equally well in high dimensions $n\ge 2$ for the energy \[ J[ρ]=\iint |x-y|^{2-n}dρ(x)dρ(y)+d^2(ρ, ρ_0). \]