Holder continuity and Hilber space

Question

Let $\Omega\subset \Re^n$ be an open set and let $u \in H^1_{loc}(\Omega)$ be a weak solution of $\Delta u=f $ in $\Omega$, with $f \in C^{0,\alpha}(\Omega)$. Prove that $u \in C^{2,\alpha}(K)$ for any compact $K \subset \Omega$.

Answer 1

This is a standard theorem in elliptic PDE: see the book by Gilbarg and Trudinger or the lecture notes Regularity for Poisson equation by Xinwei Yu. Starting from scratch, the proof takes several pages. The basic idea is to consider the Newtonian potential of $f$ (call it $v$), which is shown to be in $C^{2,\alpha}$ by direct estimates. The difference $u-v$ is harmonic in the weak sense, hence (Weyl's lemma) in the classical sense, hence $u-v\in C^\infty$.