Functions and Domains Having Minimal Resistance Under a Single-Impact Assumption
Myriam Comte, Thomas Lachand-Robert · SIAM Journal on Mathematical Analysis · 2002
We are looking for the domains $\Omega\subset\R^2$ tiling the plane and for functions $u:\Omega\to \R$ satisfying the simple impact assumption introduced by G. Buttazzo, V. Ferone, and B. Kawohl [{\it Math. Nach.}, 173 (1993), pp. 71--89.] about Newton's problem of the body of minimal resistance, which minimizes functionals $F(u;\Omega) = \frac1{\abs\Omega}\int_\Omega f(\abs{ abla u})$, with f decreasing. We prove that only some convex polygons are minimizers, and we give explicitly the corresponding functions u. In the case of the Newton's functional f(t) =1/(1+t 2 ), all optimal domains are squares or regular hexagons.