We consider the following problem: minimize the functional \(\int _\Omega f(\nabla u(x))\, dx\) in the class of concave functions \(u: \Omega \rightarrow [0,M]\), where \(\Omega \subset {\mathbb {R}}^2\) is a convex body and \(M > 0\). If \(f(x) = 1/(1 + |x|^2)\) and \(\Omega \) 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 \(\partial \Omega \) are regular and \({(1+|x|)f(x)}/(|y|f(y)) \rightarrow +\infty \) as \((1+|x|)/|y| \rightarrow 0\) then a solution u to the problem satisfies \(u\rfloor _{\partial \Omega } = 0\). 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.

我们考虑以下问题：在凹函数类别中最小化函数\（\ int _ \ Omega f（\ nabla u（x））\，dx \）\（u：\ Omega \ rightarrow [0，M] \ ），其中\（\ Omega \ subset {\ mathbb {R}} ^ 2 \）是凸体，而\（M> 0 \）。如果\（f（x）= 1 /（1 + | x | ^ 2）\）并且\（\ Omega \）是一个圆，则该问题称为最小阻力的牛顿问题。已知问题允许至少一种解决方案。我们证明如果\（\ partial \ Omega \）的所有点都是规则的，并且\（{（1+ | x |）f（x）} /（| y | f（y））\ rightarrow + \ infty \）作为\（（1+ | x |）/ | y | \ rightarrow 0 \）然后是解u问题满足\（u_rfloor _ {\ partial \ Omega} = 0 \）。这个结果证明了Buttazzo和Kawohl在1993年的论文中提出的猜想（Math Intell 15：7–12，1993）对牛顿问题的猜想。