On the Newton partially flat minimal resistance body type problems

Abstract

We study the flat region of stationary points of the functional ${\int }_{\Omega }F(\left|\nabla u(x)\right|)dx$ under the constraint $u\le M,$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^2$. Here $F(s)$ is a function which is concave for $s$ small and convex for $s$ large, and $M>0$ is a given constant. The problem generalizes the classical minimal resistance body problems considered by Newton. We construct a family of partially flat radial solutions to the associated stationary problem when $\Omega$ is a ball. We analyze also some other qualitative properties. Moreover, we show the uniqueness of a radial solution minimizing the above mentioned functional. Finally, we consider nonsymmetric domains $\Omega$ and provide sufficient conditions which insure that a stationary solution has a flat part.