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## Weakly harmonic implies harmonic

Suppose $u \in C^0(\Omega)$ is weakly harmonic on an open set $\Omega$, i.e. the relation $\displaystyle \int_\Omega u \Delta \phi dx =0$ holds for all $\phi \in C_0^2(\Omega)$. Show that $u$ is harmonic in $\Omega$.

PHD Iowa (6202)

See this blog post for a proof.