Method of nose stretching in Newton's problem of minimal resistance

We consider the problem inf{integral integral(Omega)(1+| del u(x(1),x(2))(|2))(-1)dx(1)dx(2):thefunction u:Omega -> R isconcave and 0 <= u(x)<= M for all x=(x(1),x(2))is an element of Omega={|x|<= 1} (Newton's problem) and its generalizations. In the paper by Brock, Ferone, and Kawohl (1996) it is proved that if a solution u is C (2) in an open set U subset of Omega D (2) u = 0 in U (u)U u. In this paper we prove a somewhat stronger result. Namely, there exists a solution u possessing the following property. If u is C (1) in an open set U subset of omega (uU) C (u) = {(x, z): x is an element of omega, 0 <= z <= u(x)}. As a consequence, we have Cu=Conv(SingCu), where SingC (u) denotes the set of singular points of partial derivative C (u) . We prove a similar result for a generalization of Newton's problem.