W^{2,1} REGULARITY FOR SOLUTIONS of Monge-Ampere equation

Question

In the paper by GUIDO DE PHILIPPIS AND ALESSIO FIGALLI:
http://arxiv.org/abs/1111.7207

They proved the $W^{2,1}$ estimate for standard monge-ampere equation $detD^{2}u=f$ with $f$ bounded from below and above. But there is one part I can not understand very well. In page 5, at the beginning of section 3, they said "By standard approximation arguments, it suffices to prove (1.2) when $u\in C^{2}$"

I guess what they mean is molify $f$, namely $f_{h}$, solve $detD^{2}u_{h}=f_{h}$, then use $u_{h}$ to approximate $u$. But the problem is when you try to prove $u_{h}$ has a subsequence converges in $W^{2,1}$, while one can see $u_{h}-u$ only satisfies an equation which is not uniformly elliptic, so there should be some problem for this kind of approximation?

I feel like: we assume $u$ is $W^{2,1}$, and then we prove some $W^{2,1}$ estimates. this is the so called a priori estimate?

But for monge ampere equation, I can not see this apriori estimate implies the solution is $W^{2,1}$

Answer 1

One possible approximation argument is spelt out in detail in section 5 of this paper of Schmidt.

There the author proves the stronger result that under your same assumption $0<\lambda\leq f\leq \Lambda$ then $u\in W^{2,1+\varepsilon}_{\rm loc}$ for some $\varepsilon(n,\lambda,\Lambda)>0$. This was also proved at the same time by De Philippis, Figalli and Savin.