Gradient estimate and $L^1$ theory for the Laplace operator

Question

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.

Answer 1

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 $u \in C^2$.