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

Take a nonnegative radial mollifier ${\eta }_{\epsilon }$ supported in $B_{\epsilon }(0)$, with $\epsilon <\mathrm{dist}(x,\partial \Omega )$. Writing the convolution in polar coordinates and applying the spherical mean value property for harmonic functions at every radius shows directly thatThe convolution is smooth, so $u$ agrees locally with a smooth function. Since $x$ was arbitrary, $u\in C^{\infty }(\Omega )$.