边界上Dirichlet区域输运密度的对称化求和估计
Summability estimates on transport densities with dirichlet regions on the boundary via symmetrization techniques
摘要：
本文考虑有界区域$Ω$中的质量运输问题，其中内部的正质量f +被发送到边界$\partialΩ$，例如出现在一些形状最优化问题中，并且我们证明了相关的输运密度$σ$的可和性估计，它是从扩散测度到边界上测度的输运密度f -- = P f +（P是边界上的投影），因此是奇异的。通过对称化技巧，只要$Ω$是凸的或满足一致的外球条件，我们就证明L p估计（如果f + $\in$ L p，则$σ$ $\in$ L p）。最后，通过反例证明了如果f + $\in$ L $\infty$ $(Ω)$和f --有界密度w.r.在$\partialΩ$上的表面测量，f +和f -之间的输运密度$σ$不一定在L $\infty$ $(Ω)$中，这意味着f - = P \# f +的事实是至关重要的。
英文摘要：
In this paper we consider the mass transportation problem in a bounded domain $Ω$ where a positive mass f + in the interior is sent to the boundary $\partialΩ$, appearing for instance in some shape optimization problems, and we prove summability estimates on the associated transport density $σ$, which is the transport density from a diffuse measure to a measure on the boundary f -- = P \# f + (P being the projection on the boundary), hence singular. Via a symmetrization trick, as soon as $Ω$ is convex or satisfies a uniform exterior ball condition, we prove L p estimates (if f + $\in$ L p, then $σ$ $\in$ L p). Finally, by a counterexample we prove that if f + $\in$ L $\infty$ $(Ω)$ and f -- has bounded density w.r.t. the surface measure on $\partialΩ$, the transport density $σ$ between f + and f -- is not necessarily in L $\infty$ $(Ω)$, which means that the fact that f -- = P \# f + is crucial.
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