# Schauder regularity

My question is: If $$\Omega$$ is bounded with smooth boundary...

Let $$f \in C^{0,\alpha}(\overline{\Omega})$$, and let $$u$$ be a weak solution of the Poisson equation $$-\Delta u = f$$. Then $$u \in C^{2,\alpha}(\overline{\Omega})$$.

I know that there exists Schauder interior estimates, boundary estimates, etc. But in those theorems we asume $$u \in C^{2}(\Omega)$$. How can we prove it when $$u$$ is only a weak solution?