We consider the problem of finding two free export/import sets $E^+$ and $E^-$ that minimize the total cost of some export/import transportation problem (with export/import taxes $g^\pm $ ), between two densities $f^+$ and $f^-$ , plus penalization terms on $E^+$ and $E^-$ . First, we prove the existence of such optimal sets under some assumptions on $f^\pm $ and $g^\pm $ . Then we study some properties of these sets such as convexity and regularity. In particular, we show that the optimal free export (resp. import) region $E^+$ (resp. $E^-$ ) has a boundary of class $C^2$ as soon as $f^+$ (resp. $f^-$ ) is continuous and $\partial E^+$ (resp. $\partial E^-$ ) is $C^{2,1}$ provided that $f^+$ (resp. $f^-$ ) is Lipschitz.

我们考虑找到两个自由出口/进口集 $E^+$ 和 $E^-$ 的问题 ，它们使某些出口/进口运输问题的总成本最小化（出口/进口税 为 $g^\pm $ ），介于两个密度 $f^+$ 和 $f^-$ ，加上 $E^+$ 和 $E^-$ 的 惩罚项。首先，我们在 $f^\pm $ 和 $g^\pm $ 的 一些假设下证明了这样的最优集的存在。然后我们研究这些集合的一些性质，例如凸性和正则性。特别地，我们证明了最佳自由出口（相应的进口）区域 $E^+$ （相应的 $E^-$ )只要 $f^+$ (resp. $f^-$ ) 是连续的并且 $\partial E^+$ (resp. $\partial E^-$ ) 就具有类 $C^2$ 的边界 $C^{2,1}$ 前提是 $f^+$ (resp. $f^-$ ) 是 Lipschitz。