A nonlocal free boundary problem with Wasserstein distance

Abstract

We study the probability measures $ρ\in \mathcal M(\mathbb R^2)$ minimizing the functional \[ J[ρ]=\iint \log\frac1{|x-y|}dρ(x)dρ(y)+d^2(ρ, ρ_0), \] where $ρ_0$ is a given probability measure and $d(ρ, ρ_0)$ is the 2-Wasserstein distance of $ρ$ and $ρ_0$. % We prove the existence of minimizers $ρ$ and show that the potential $U^ρ=-\log|x|\ast ρ$ solves a degenerate obstacle problem, the obstacle being the transport potential. Every minimizer $ρ$ 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, $U^ρ$ solves a nonlocal Monge-Ampére equation, which after linearization leads to the equation $ρ_t={\hbox{div}}(ρ\nabla U^ρ)$. The methods we develop use Fourier transform techniques. They work equally well in high dimensions $n\ge2$ for the energy \[ J[ρ]=\iint |x-y|^{2-n}dρ(x)dρ(y)+d^2(ρ, ρ_0). \]