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Optimal free export/import regions

Part of: Operations research and management science Manifolds

Published online by Cambridge University Press:  17 September 2020

Samer Dweik
Samer Dweik*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
*
e-mail: dweik@math.ubc.ca
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Abstract

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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.

Keywords

Shape optimizationoptimal regionsimport/export transport problem

MSC classification

Primary: 49Q10: Optimization of shapes other than minimal surfaces
Secondary: 49Q20: Variational problems in a geometric measure-theoretic setting 90B20: Traffic problems

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 737 - 751
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Copyright
© Canadian Mathematical Society 2020

References

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