Two-Dimensional Problems of Minimal Resistance in a Medium of Positive Temperature

Two-dimensional body of maximum mean resistance

A two-dimensional body, exhibiting a slight rotational movement, moves in a rarefied medium of particles which collide with it in a perfectly elastic way. In previously realized investigations by the first two authors, Plakhov & Gouveia (2007, Nonlinearity, 20), shapes of nonconvex bodies were sought which would maximize the braking force of the medium on their movement. Giving continuity to this study, new investigations have been undertaken which culminate in an outcome which represents a large qualitative advance relative to that which was achieved earlier. This result, now presented, consi…

A note on Newton's problem of minimal resistance for convex bodies

We consider the following problem: minimize the functional $\int_Ωf(\nabla u(x))\, dx$ in the class of concave functions $u: Ω\to [0,M]$, where $Ω\subset \mathbb{R}^2$ is a convex body and $M > 0$. If $f(x) = 1/(1 + |x|^2)$ and $Ω$ is a circle, the problem is called Newton's problem of least resistance. It is known that the problem admits at least one solution. We prove that if all points of $\partialΩ$ are regular and ${|x|f(x)}/(|y|f(y)) \to +\infty$ as $|x|/|y| \to 0$ then a solution $u$ to the problem satisfies $u\rfloor_{\partialΩ} = 0$. This result proves the conjecture stated in 1993 fo…

最小阻力问题，旧与新

The problem of minimal resistance, old and new

本文综述了变分法中最小阻力问题的历史进展与理论成果，并提出了未来的新研究方向。

Billiards and two-dimensional problems of optimal resistance

A body moves in a medium composed of noninteracting point particles; interaction of particles with the body is absolutely elastic. It is required to find the body's shape minimizing or maximizing resistance of the medium to its motion. This is the general setting of optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary $B \subset \mathbb{R}^2$ we assign a measure $ν_B$ on $\partial(…

Two-dimensional Newton's Problem of Minimal Resistance

Newton's problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in dimension three, the shape of an axis-symmetric body, with assigned radius and height, which offers minimum resistance when it is moving in a resistant medium. The problem has a very rich history and is well documented in the literature. Of course, at first glance, one suspects that the two dimensional case should be well known. Nevertheless, we have looked into n…

一种用于选择最小制造成本和所需加权力-阻力性能的双弹簧晶格-弹簧拓扑的降维程序

A dimension reduction procedure for the selection of Two-spring lattice-spring topologies with minimal fabrication cost and required weighted force-resistance performance

本文提出了一种降维方法，用于选择在最小制造成本和所需加权力-阻力性能下最优的双弹簧晶格-弹簧拓扑结构。

Method of nose stretching in Newton's problem of minimal resistance

We consider the problem $\inf\big\{ \int\!\!\int_Ω(1 + |\nabla u(x,y)|^2)^{-1} dx dy : \text{ the function } u : Ω\to \mathbb{R} \text{ is concave and } 0 \le u(x,y) \le M \text{ for all } (x,y) \in Ω=\{ (x,y): x^2 + y^2 \le 1 \} \, \big\}$ (Newton's problem) and its generalizations. In the paper \cite{BrFK} it is proved that if a solution $u$ is $C^2$ in an open set $\mathcal{U} \subset Ω$ then $\det D^2u = 0$ in $\mathcal{U}$. It follows that graph$(u)\rfloor_\mathcal{U}$ does not contain extreme points of the subgraph of $u$. In this paper we prove a somewhat stronger result. Namely, there …

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

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。

Spinning rough disk moving in a rarefied medium

A spinning rough disk moves through a rarefied medium on the plane. The roughness is formed by small cavities on the disk boundary. The medium is so rare that mutual interaction of particles can be neglected. All collisions of particles with the disk are perfectly elastic; there may happen multiple collisions in the cavities. We calculate the force of resistance acting on the body and examine how it depends on the kind of roughness (shape of the cavities). We show that the nonzero transversal component of the force generally appears, resulting in deflection of the disk trajectory. In several s…

The analytical solution to Newton's aerodynamic problem in the class of bodies with vertical plane of symmetry and developable side boundary

The method of Hessian measures is used to find the differential equation that defines the optimal shape of nonrotationally symmetric bodies with minimal resistance moving in a rare medium. The synthesis of optimal solutions is described. A theorem on the optimality of the obtained solutions is proved.