Regularity of a function between two paraboloids tangents I know that the regularity of a continuous function $u$ between two paraboloids tangents in a neighbourhood of a point $x_0$ is $C^{1,1}$. I'd like to see for example, how to prove that $u$ is differentiable at $x_0$.
 A: By assumption,  the two paraboloids are tangent to each other at $x_0$. Let $z=z_0+\langle a,x-x_0\rangle$ be the equation of their common tangent plane. Then the equations of paraboloids, $z=f_i(x)$, $i=1,2$, satisfy 
$$\lim_{x\to x_0} \frac{|f_i(x) - z_0 - \langle a,x-x_0\rangle|}{ |x-x_0|}=0 $$
If $u$ is squeezed between paraboloids,  $f_1\le u\le f_2$, then  $$\lim_{x\to x_0} \frac{|u(x) - z_0 - \langle a,x-x_0\rangle|}{ |x-x_0|}=0 $$
which means that $u$ is differentiable at $x_0$. 
Note that the specific shape of paraboloids did not matter. Squeezing $u$ between any two differentiable functions with common tangent plane at $x_0$ guarantees that $u$ is differentiable at $x_0$. 

To say that $u\in C^{1,1}$ we need such a squeeze at other points too (with Lipschitz dependency of $a$ on the point of tangency). Having it only at $x_0$ would not be enough: for example, in one dimension
$$u(x) =   \chi_\mathbb Q (x)x^2   $$
is squeezed between $x^2$ and $-x^2$, but is not continuous anywhere except $0$. 
A: The $C^{1,1}$ regularity requires a more subtle argument. Actually (to the best of my knowledge) you need to be in a convex subset compactly contain in the domain in order to be able to develop the argument. Also you need a uniform boundedness in your convex set of the function
\begin{equation}
\Theta(u,\epsilon)(x)=\sup\{\overline\Theta(u,\epsilon)(x),\underline\Theta(u,\epsilon)(x)\}.
\end{equation} 
$\overline\Theta(u,\epsilon)(x)$ [resp. $\underline\Theta(u,\epsilon)(x)$] is the infimum over the positive $M$ such that there exists a paraboloid 
\begin{equation}
P(x)=L(x)+\frac{M}{2}\vert x\vert^2 \quad[\textrm{resp. }P(x)=L(x)-\frac{M}{2}\vert x\vert^2]
\end{equation} 
such that $P(x)\geq u(x)$ [resp. $P(x)\leq u(x)$] in $\Omega\cap B_{\epsilon}(x)$ for $x\in\overline B$ (here $L(x)$ is an affine function). And on the top of it all that, you have to get a result saying that you can control the $L^p$ norm of the Hessian of $u$ by the corresponding norm of $\Theta(u,\epsilon)(x)$. You can find the proof of both results in Caffarelli and Cabre's book on fully nonlinear equations.
