A note on Newton's problem of minimal resistance for convex bodies

Abstract

We consider the following problem: minimize the functional f (∇u(x)) dx in the class of concave functions u : D → [0, M], where D ⊂ R2 is a convex body and M > 0. If f (x) = 1/(1 + |x|^2) and D is a circle, the problem is called Newton’s problem of least resistance. It is known that the problem admits at least one solution. We prove that if all points of ∂D are regular and (1 + |x|) f (x)/(|y| f (y)) → +∞ as (1 + |x|)/|y| → 0 then a solution u to the problem satisfies u|_∂D = 0. This result proves the conjecture stated in 1993 in the paper by Buttazzo and Kawohl (Math Intell 15:7–12, 1993) for Newton’s problem.