Let $B$ be the unit ball in $\mathbb{R}^{n}$ and $u\in W^{2,p}(B)$, with $p>\dfrac{n}{2}$. How can we see that $u$ is second differentiable almost everywhere in $B$?
This result is claimed in Page 25 but I cannot prove it. I can understand the proof of: $v\in W^{1,p}(B)$, with $p>n$ then $v$ is differentiable almost everywhere in $B$ (in the same page of the link).
My attempt so far:
I try to follow the same method as in Page 25.
Assume that $0$ is a Lebesgue point of $D^2u\in L^p$, i.e.
$$ |B_r|^{-1}\int_{B_r}|D^2u(x)-D^2u(0)|^p\rightarrow 0, \text{as }r\rightarrow 0. \quad (*)
$$
Here $B_r$ denotes the ball of radius $r$ centered at the origin. Now, our aim is to show that $u$ is classically second diffentiable at $0$. Set
$$
h(x)= u(x)-u(0)-Du(0)x-D^2u(0)\dfrac{x^2}{2},
$$
and set $h_r(x):=h(rx)/r^{2}$. It is clear to see that it suffices to show
$$
||h_r||_{L^{\infty}(B)}\rightarrow 0, \text{as }r\rightarrow 0.\quad \textbf{(1)}
$$
Since $u\in W^{2,p}(B)$, with $p>\dfrac{n}{2}$, it is not hard to see that $u\in W^{1,q}(B)$, for some $q>n$. For example, assume for simplicity that $n>p>\frac{n}{2}$ then we can choose $q=np/(n-p)$.
By Morrey's inequality (since $q>n$), $$ ||h_r-h_r(0)||_{L^{\infty}(B)}\leq C ||Dh_r||_{L^q(B)}. $$
Note that $h_r(0)=0$, and (*) is equivalent to $||D^2h_r||_{L^p(B)}\rightarrow 0$. Therefore, in order to finish the proof (i.e. to show $\textbf{(1)}$ is true), we really hope to have the following inequality $$ ||Dh_r||_{L^{q}(B)}\leq C ||D^{2}h_{r}||_{L^{p}(B)}. $$ However, this cannot be seen by the Sobolev embedding theorem since the Sobolev embedding on a bounded domain should be $$ ||Dh_r-|B|^{-1}\int_{B}Dh_r||_{L^{q}(B)}\leq C ||D^{2}h_{r}||_{L^{p}(B)}. $$
How can I resolve this problem?
Thanks for any suggestion.
Hint: From $$ \left\|Dh_r-|B|^{-1}\int_{B}Dh_r\right\|_{L^{q}(B)}\leq C \|D^{2}h_{r}\|_{L^{p}(B)}. $$ and the triangle inequality, you have $$ \left\|Dh_r\right\|_{L^{q}(B)}\leq C \|D^{2}h_{r}\|_{L^{p}(B)} + \hat C \, \left|\int_{B}Dh_r\right|. $$