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- W4387228807 abstract "We study the probability measures $rhoin mathcal M(mathbb R^2)$ minimizing the functional [ J[rho]=iint logfrac1{|x-y|}drho(x)drho(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 $rho_t={hbox{div}}(rhonabla U^rho)$. The methods we develop use Fourier transform techniques. They work equally well in high dimensions $nge2$ for the energy [ J[rho]=iint |x-y|^{2-n}drho(x)drho(y)+d^2(rho, rho_0). ]" @default.
- W4387228807 created "2023-10-01" @default.
- W4387228807 creator A5023199868 @default.
- W4387228807 date "2019-04-12" @default.
- W4387228807 modified "2023-10-16" @default.
- W4387228807 title "A nonlocal free boundary problem with Wasserstein distance" @default.
- W4387228807 doi "https://doi.org/10.48550/arxiv.1904.06270" @default.
- W4387228807 hasPublicationYear "2019" @default.
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