 Solution

For each $0<r<1$, the spherical mean value property for harmonic functions givesInsert this into part 1(iii). The normalization of the radial mollifier is $d{\omega }_d{\int }_0^1\varphi (r)r^{d-1}dr=1$, henceA convolution with a smooth function of compact support is smooth wherever it is defined. Every point admits such an $\epsilon$, so $u\in C^{\infty }(\Omega )$.