A nonlocal free boundary problem with Wasserstein distance
A nonlocal free boundary problem with Wasserstein distance
复制标题
具有 Wasserstein 距离的非局部自由边界问题
DOI：
10.1007/s00526-023-02581-9
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发表时间：
2023
影响因子：
2.1
通讯作者：
Karakhanyan A
中科院分区：
文献类型：
--
作者：
Karakhanyan A
We study the probability measuresminimizing the functional \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}whereis a given probability measure andis the 2-Wasserstein distance ofand.appears in aggregation models when the movement of particles is advanced by the potential. We prove the existence of minimizersand show that the potentialsolves a degenerateobstacle problem, the obstacle being the transport potential. Every minimizeris 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.Moreover,solves a nonlocal Monge–Ampère equation, which after linearization leads to the equation. The methods we develop use Fourier transform techniques. They work equally well in high dimensionsfor the energy \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}
We study the probability measuresminimizing the functional \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}whereis a given probability measure andis the 2-Wasserstein distance ofand.appears in aggregation models when the movement of particles is advanced by the potential. We prove the existence of minimizersand show that the potentialsolves a degenerateobstacle problem, the obstacle being the transport potential. Every minimizeris 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.Moreover,solves a nonlocal Monge–Ampère equation, which after linearization leads to the equation. The methods we develop use Fourier transform techniques. They work equally well in high dimensionsfor the energy \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}
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DOI：
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发表时间：
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期刊：
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影响因子：
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