Normal mappings in Alexandrov Pucci maximum principle

I'm reading Chen and Wu's book, Second order elliptic equations and elliptic systems. In the chapter on Krylov and Safonov theorem for nondivergence form elliptic equations, the concept of normal mapping is introduced. On page 81, definition 1.4 it is said that if $\Omega \subset \mathbf{R}^n$ is a domain, $x_0 \in \Omega$, for $\lambda \in \mathbf{R}$ they set $w$ the function such that its graph is a cone surface with vertex at $(x_0, \lambda)$ and base $\Omega$.

I tried to construct such a function, but failed. For example if one takes $\Omega \subset \mathbf{R}^2$ to be the domain which boundary is: $B\ =\ A_1 \cup A_2 \cup A_3 \cup A_4 \cup A_5 \cup A_6$ with:

1. $A_1\ =\ \{(x,0): 1 \le x \le 2\}$
2. $A_2\ =\ \{(1, y): 0 \le y \le 1\}$
3. $A_3\ =\ \{(2,y): -1 \le y \le 0\}$
4. $A_4\ =\ \{(x,-1): -1 \le x \le 2\}$
5. $A_5\ =\ \{(-1, y): -1 \le y \le 1\}$
6. $A_6\ =\ \{(x, 1): -1 \le x \le 1\}$

Then if we take $x_0\ =\ (0,0)$ and $\lambda =1$ I don't understand how to construct the function whose graph is a cone with vertex in $(x_0, 1)$ and base $\Omega$. In fact every point $(x,0)$ with $x > 0$ has "lots of lines" above it. Shall I assume $\Omega$ convex or am I doing something wrong?

• I think you're right, but it's not important - if I remember correctly the function $w$ itself is not used, only the image of the normal mapping of the cone, which is well-defined whether or not the cone happens to be a graph. – Anthony Carapetis Nov 11 '17 at 11:18
• Not really. It uses lemma 1.2 on the normal mapping to say that the set $\Omega [x_0, \lambda]$ is measurble. So he is assuming also the continuity of the function. Anyway, how do you define the normal mapping without the function? Do you have any reference? Thank you @Anthony Carapetis – jJjjJ Nov 11 '17 at 11:25
• $\chi_u(y)$ is just the normal cone (in the sense of convex analysis) to the undergraph of $u$ at $(y,u(y))$, so you can generalize this. When $w$ is "multivalued" you may need to take the union of normal cones at the different vertical positions, or maybe just the top one; I'm not sure. As to the regularity, the cone should always be a graph locally, so this should be fine; though of course there is something to prove here. I've seen this proof before in chap 9 of Gilbarg-Trudinger but they made the same mistake(?)/omission there, so I'm not sure if it would be useful. – Anthony Carapetis Nov 11 '17 at 12:18
• Perhaps for the purposes of obtaining the estimates you can just replace $\Omega$ with a ball containing it? This is the approach taken by Caffarelli and Cabré in Fully nonlinear elliptic equations, though they are working with viscosity solutions. – Anthony Carapetis Nov 11 '17 at 12:20
• It might help to look at some of the original papers, though unfortunately it seems hard to find English translations online. For example in Pucci's Limitazioni per soluzioni di equazioni elIittiche it looks like $\Omega[x_0,\lambda]$ is defined using the convex hull of $\Omega$. – Anthony Carapetis Nov 11 '17 at 12:28

Hessian could be looked as a Jacobi metrix of $(u_1,...,u_n)\to (e_1,...,e_n)$. Elliptic property is just said the graph of the function could not have very narrow cone. By area formula and a rescaling we will derive ABP estimate.