Problems of optimal transportation on the circle and their mechanical applications

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• 作者：Alexander Plakhov^a ; ^b ; ^{plakhov@ua.pt} ; Tatiana Tchemisova^a
• 关键词：49Q10 ; 49K30 ; 37N05
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：262
• 期：3
• 页码：2449-2492
• 全文大小：1054 K
• 卷排序：262

文摘

We consider a mechanical problem concerning a 2D axisymmetric body moving forward on the plane and making slow turns of fixed magnitude about its axis of symmetry. The body moves through a medium of non-interacting particles at rest, and collisions of particles with the body's boundary are perfectly elastic (billiard-like). The body has a blunt nose: a line segment orthogonal to the symmetry axis. It is required to make small cavities with special shape on the nose so as to minimize its aerodynamic resistance. This problem of optimizing the shape of the cavities amounts to a special case of the optimal mass transportation problem on the circle with the transportation cost being the squared Euclidean distance. We find the explicit solution for this problem when the amplitude of rotation is smaller than a fixed critical value, and give a numerical solution otherwise. As a by-product, we get explicit description of the solution for a class of optimal transportation problems on the circle.