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.

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.