= Harmonic functions are smooth
Every $C^2$ <harmonic function> on an <open set> in $\mathbb R^n$ is infinitely differentiable, in every dimension. Let $\rho_\varepsilon$ be a smooth radial <mollifier> supported in a radius-$\varepsilon$ ball, with integral1. Radially integrating the <mean value property for harmonic functions> gives
$$
u*\rho_\varepsilon=u
$$
wherever that ball stays inside the domain. The left side is smooth, so $u$ is smooth locally.<Partial Differential Equations by John K. Hunter>[chapter2, Theorem2.2 and its mollification proof, printed p21]{ref}
In two dimensions there is also a <complex analysis> proof: locally $u$ is the <real part> of a <holomorphic function>, whose <power series> can be differentiated to every order. That proof is planar; it is not a justification for arbitrary $n$. More generally, the <Weyl lemma> supplies a smooth representative even when the initial equation $\Delta u=0$ holds only in <distributions>.
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