Elliptic regularity estimate for Possion with bounded forcing term

Question

Suppose I have $- \Delta u = f $ weakly on a bounded domain $\Omega$ with Dirichlet boundary conditions. I'm looking for an elliptic regularity result of the form $f \in L^\infty \implies u \in C^{1, \alpha}(\Omega)$. The usual Schauder/Newtonian potential results I know require $f \in C^{\alpha}(\Omega)$ but give $u \in C^{2, \alpha}(\Omega)$ Is there a result of the type I am looking for?

Answer 1

Since $\Omega$ is bounded, you have that $f\in L^p$ for $p>N$. Assuming the boundary is sufficiently regular and that the Dirichlet boundary condition is zero, you can apply Sobolev regularity, which tells you that $u$ is in $W^{2,p}$. Hence its gradient is in $W^{1,p}$. Now you can apply Morrey’s embedding theorem which says that a Sobolev function in $W^{1,p}$ for $p>N$ is Holder continuous (you need the boundary to be smooth). You can find the details in Gilbarg and Trudinger “ Second order elliptic equations”.