Calculus of Variations and Geometric Measure Theory

S. Dweik

Optimal free export/import regions

created by dweik on 25 Oct 2020


Accepted Paper

Inserted: 25 oct 2020
Last Updated: 25 oct 2020

Journal: Canadian Mathematical Bulletin
Year: 2020


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 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 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.