On generalized Newton’s aerodynamic problem

We consider the generalized Newton’s least resistance problem for convex bodies: minimize the functional ∬ Ω ( 1 + | ∇ u ( x , y ) | 2 ) − 1 d x d y \iint _\Omega (1 + |\nabla u(x,y)|^2)^{-1} dx\, dy in the class of concave functions u : Ω → [ 0 , M ] u\colon \Omega \to [0,M] , where the domain Ω ⊂ R 2 \Omega \subset \mathbb {R}^2 is convex and bounded and M > 0 M > 0 . It has been known (see G. Buttazzo, V. Ferone, and B. Kawohl [Math. Nachr. 173 (1995), pp. 71–89]) that if u u solves the problem, then | ∇ u ( x , y ) | ≥ 1 |\nabla u(x,y)| \ge 1 at all regular points ( x , y ) (x,y) such that u ( x , y ) > M u(x,y) > M . We prove that if the upper level set L = { ( x , y ) : u ( x , y ) = M } L = \{ (x,y)\colon u(x,y) = M \} has nonempty interior, then for almost all points of its boundary ( x ¯ , y ¯ ) ∈ ∂ L (\bar {x}, \bar {y}) \in \partial L one has lim ( x , y ) → ( x ¯ , y ¯ )