JournalsjemsVol. 7 , No. 4DOI 10.4171/jems/33

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
On the Newton partially flat minimal resistance body type problems cover

Abstract

We study the flat region of stationary points of the functional ΩF(u(x))dx\int_{\Omega }F(\left\vert \nabla u(x)\right\vert )dx under the constraint uM,u\leq M, where Ω\Omega is a bounded domain of R2{\mathbb{R}}^{2}. Here F(s)F(s) is a function which is concave for ss small and convex for ss large, and M>0M>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 Ω\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 Ω\Omega and provide sufficient conditions which insure that a stationary solution has a flat part.