Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 1 / c / Solution

For $u\in L_{\mathrm{loc}}^2(\Omega )$, the correct formulation is the distributional identityThus $\Delta u=0$ as a distribution. The Weyl lemma applies already to locally integrable distributions, so $u$ agrees almost everywhere with a smooth harmonic function.