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Optimal free export/import regions

Published online by Cambridge University Press:  17 September 2020

Samer Dweik*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

Abstract

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.

Type
Article
Copyright
© Canadian Mathematical Society 2020

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