Functions and Domains Having Minimal Resistance Under a Single-Impact Assumption

We are looking for the domains Ω⊂² tiling the plane and for functions u:Ω→ satisfying the simple impact assumption introduced by G. Buttazzo, V. Ferone, and B. Kawohl [{ Math. Nach.}, 173 (1993), pp. 71--89.] about Newton's problem of the body of minimal resistance, which minimizes functionals F(u;Ω) = 1Ω∫_Ω f(∇ 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.