# On the Newton partially flat minimal resistance body type problems

• Volume: 007, Issue: 4, page 395-411
• ISSN: 1435-9855

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## Abstract

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We study the flat region of stationary points of the functional ${\int }_{\Omega }F\left(|\nabla u\left(x\right)|\right)dx$ under the constraint $u\le M$, where $\Omega$ is a bounded domain in ${ℝ}^{2}$. Here $F\left(s\right)$ 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 also analyze 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 ensure that a stationary solution has a flat part.

## How to cite

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Comte, M., and Díaz, Jesus Ildefonso. "On the Newton partially flat minimal resistance body type problems." Journal of the European Mathematical Society 007.4 (2005): 395-411. <http://eudml.org/doc/277690>.

@article{Comte2005,
abstract = {We study the flat region of stationary points of the functional $\int _\Omega F(|\nabla u(x)|)dx$ under the constraint $u\le M$, where $\Omega$ is a bounded domain in $\mathbb \{R\}^2$. 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 also analyze 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 ensure that a stationary solution has a flat part.},
author = {Comte, M., Díaz, Jesus Ildefonso},
journal = {Journal of the European Mathematical Society},
keywords = {stationary points; minimal resistance body problems; Newton problem; obstacle problem; flat solutions; quasilinear elliptic operators; quasilinear elliptic operators; Newton problem; obstacle problem; aerodynamical minimal resistance},
language = {eng},
number = {4},
pages = {395-411},
publisher = {European Mathematical Society Publishing House},
title = {On the Newton partially flat minimal resistance body type problems},
url = {http://eudml.org/doc/277690},
volume = {007},
year = {2005},
}

TY - JOUR
AU - Comte, M.
AU - Díaz, Jesus Ildefonso
TI - On the Newton partially flat minimal resistance body type problems
JO - Journal of the European Mathematical Society
PY - 2005
PB - European Mathematical Society Publishing House
VL - 007
IS - 4
SP - 395
EP - 411
AB - We study the flat region of stationary points of the functional $\int _\Omega F(|\nabla u(x)|)dx$ under the constraint $u\le M$, where $\Omega$ is a bounded domain in $\mathbb {R}^2$. 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 also analyze 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 ensure that a stationary solution has a flat part.
LA - eng
KW - stationary points; minimal resistance body problems; Newton problem; obstacle problem; flat solutions; quasilinear elliptic operators; quasilinear elliptic operators; Newton problem; obstacle problem; aerodynamical minimal resistance
UR - http://eudml.org/doc/277690
ER -

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