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

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.

One way to get both is to use the estimates $$\|\phi\|_{W^{2,p}(\Omega)} \leq C\|\psi\|_{L^p(\Omega)}$$ which hold when $$1 with a constant $$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$$.