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Second derivative estimates

I am in big trouble since I don’t see how to proceed (I don’t need the exact calculation) with the following estimates.

In one of his papers, Lin proves the following result:

Let’s consider a bounded, smooth domain $\Omega$ in $\mathbb{R}^n$ and let $(a_{ij})$ be a symmetric $n$x$n$ matrix-valued function on $\Omega$ wich satisfy $\lambda I \leq a(x) \leq \lambda^{-1}I$. Put
$$L_a=\sum_{ij}a_{ij}(x)\dfrac{\partial^2}{\partial x_i \partial x_j}$$
and let $u\in C^{1,1}(\Omega)$ be the solution of the following Dirichlet problem:
$$
\begin{cases}
L_au=-f &\text{ in }\Omega \\
u=0 &\text{ in }\partial \Omega\end{cases}
$$

where $f \in L^n(\Omega)$.

Theorem. There is a positive constant $p=p(n,\lambda)$ such that $$
\Vert D^2u\Vert_{L^p(\Omega)}\leq c(n,\lambda,p,\Omega)\Vert f\Vert_{L^n(\Omega)}
$$

for any $u\in C^{1,1}(\Omega)$, $u=0$ in $\partial \Omega$, where $f=L_a u$ and $D^2u$ is the Hessian matrix of $u$.

Now, I want to show an other statement that use this last theorem.

N.B. The definition of convex function can be found here.

I am not interested actually in the first derivative estimates (this is what Evans says in [5]) but I want just an estimate of the second derivative. So my questions are:

  • How can I use Lin’s estimate?
  • Is it an easy application of the Lin theorem or there something deeper in the Lin paper that I have to see?

Please somebody help me.

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