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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?