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?


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