Let $\psi \in C^{\infty}\_{c}(\Omega)$ where $\Omega$ is a bounded smooth domain, and $\phi$ the solution to
\begin{equation\*}
-\Delta \phi =\psi, ~\phi|\_{\partial \Omega}=0.
 \end{equation\*}
My question is how to get the following estimate :
$$\|\phi\|\_{C^1( \overline{\Omega})} \leq C \|\psi\|\_{L^{\infty}(\Omega)}.$$
The usual elliptic gradient estimate using the Bernstein method shows
$$\|\phi\|\_{C^1( \overline{\Omega})} \leq C \sup|\psi|+ C\sup|\phi|+C\sup |\nabla \psi|.$$
The problem comes from [Stable Solutions of Elliptic Partial Differential Equations](https://i.stack.imgur.com/DqdAe.png).


Another question is how to prove:


* Let $u$ be a $L^1$ weak solution of $-\Delta u= f(u)$ with zero boundary condition, if $f \in C^{\alpha}$ for some $\alpha \in (0, 1)$ and $f(u)\in L^p$ for some $p>\frac{n}{2}$, then $u$ is $C^2$, hence a classic solution.


for the definition of $L^1$ weak solution, see
[definition](https://i.stack.imgur.com/IQ5A2.png).



Gradient estimate and $L^1$ theory for the Laplace operator
One way to get both is to use the estimates $
\|\phi\|\_{W^{2,p}(\Omega)} \leq C\|\psi\|\_{L^p(\Omega)}$
which hold when $1<p<\infty$ with a constant $C=C(p,\Omega,n)$. Taking $p>n$ by Sobolev embedding $$\|\phi\|\_{C^1(\Omega)} \leq c\_1 \|\phi\|\_{W^{2,p}(\Omega)} \leq c\_2 \|\psi\|\_{L^p(\Omega)} \leq c\_3 \|\psi\|\_{L^\infty (\Omega)},$$ since $\Omega$ is bounded. The last question is similar: since $f(u) \in L^p$, then $u \in W^{2,p}$ hence in $C^\theta$ with $\theta=1-\frac{2p}{n}$ and then $f(u)$ is Holder continuous. The Schauder theory now yields that $u \in C^2$.


