论文标题
关于解决牛顿最小电阻问题的解决方案的奇异点的结构
On the structure of singular points of a solution to Newton's least resistance problem
论文作者
论文摘要
我们考虑Buttazzo和Kawohl在1993年提出的以下问题：最小化功能$ \ int \！\！\！是凸域和$ M> 0 $。它概括了经典的最小化问题，该问题最初是由I. Newton在1687年在受限制的径向函数类别中所说的。这个问题直到现在才解决。甚至对解决方案的奇异点的结构一无所知。 在本文中，首先，我们解决了一个辅助2D最小电阻问题的家族，其次，将获得的结果应用于我们原始问题的解决方案的奇异点。更确切地说，我们得出了一个点的必要条件，即溶液的脊单数点，尤其是证明所有山脊具有水平边缘的奇异点位于顶级和零级集合。
We consider the following problem stated in 1993 by Buttazzo and Kawohl: minimize the functional $\int\!\!\int_Ω(1 + |\nabla u(x,y)|^2)^{-1} dx\, dy$ in the class of concave functions $u: Ω\to [0,M]$, where $Ω\subset \mathbb{R}^2$ is a convex domain and $M > 0$. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resistance problems and, second, apply the obtained results to study singular points of a solution to our original problem. More precisely, we derive a necessary condition for a point being a ridge singular point of a solution and prove, in particular, that all ridge singular points with horizontal edge lie on the top level and zero level sets.
