# On the Newton partially flat minimal resistance body type problems

### Jesús Ildefonso Díaz

Universidad Complutense de Madrid, Spain### M. Comte

Université Pierre et Marie Curie, Paris , France

## Abstract

We study the flat region of stationary points of the functional $\int_{\Omega }F(\left\vert \nabla u(x)\right\vert )dx$ under the constraint $u\leq M,$ 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.

## Cite this article

Jesús Ildefonso Díaz, M. Comte, On the Newton partially flat minimal resistance body type problems. J. Eur. Math. Soc. 7 (2005), no. 4, pp. 395–411

DOI 10.4171/JEMS/33