拟芬斯勒度量的等周问题与最小空气动力学阻力的形状
Isoperimetric problem for quasi-Finsler metrics and shapes of least aerodynamic resistance
- Steklov Mathematical Institute of RAS（俄罗斯科学院斯捷克洛夫数学研究所）
- HSE University（高等经济学院）
- Center for R&D in Mathematics and Applications, Department of Mathematics, University of Aveiro, Portugal（阿威罗大学数学系数学研发与应用中心）
- Institute for Information Transmission Problems, Moscow（莫斯科信息传输问题研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对牛顿空气动力学中固定面积二维凸体的最小阻力问题，将其归约为拟芬斯勒度量等周问题，引入振荡微小性准则，揭示最优物体边界奇点条件并构造均匀振荡下的精确最优形状。
AI中文摘要：
我们在牛顿空气动力学框架下考虑如下模型：平面上的稀薄介质中，一个二维凸体向前运动并缓慢振荡，振荡规律已给定。问题是求固定面积且阻力最小的物体。我们通过将其归约为拟芬斯勒度量的等周问题来解决该问题。进一步，我们引入两种振荡微小性准则。我们证明，仅当前者（后者）准则满足时，最优物体边界的前部（后部）存在奇点，其余边界光滑。最后，我们在均匀振荡情形下找到精确最优形状，并明确构造出若干形状。
英文摘要：
We consider the following model in the framework of Newtonian aerodynamics: a 2D convex body moves forward and slowly oscillates in a rarefied medium on the plane. The law of oscillations is given. The problem is to find a body of fixed area that has the smallest resistance. We solve this problem by reducing it to an isoperimetric problem for quasi-Finsler metrics. Further, we introduce two criteria of smallness of oscillations. We show that the optimal body has singularity at the front (back) part of its boundary iff the former (latter) criterion is satisfied. The rest of the boundary is smooth. Finally, we find exact optimal shapes in the case of uniform oscillations and construct several shapes explicitly.
