Regularity of a function between two paraboloids tangents

Question

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 $x_0$.

Answer 1

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 $x_0$.

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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 $0$.

Answer 2

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.