Rotational commutator for the Laplacian
= Rotational commutator for the Laplacian
{title2=$[\Delta,x_i\partial_j-x_j\partial_i]=0$}
The infinitesimal rotation $R_{ij}=x_i\partial_j-x_j\partial_i$ commutes with the <Laplacian>: the extra terms $2\partial_i\partial_j$ cancel. It is tangent to every sphere, so it preserves zero Dirichlet traces on spherical shells. If $\Delta u=f$ and $f$ is radial, uniqueness applied to $R_{ij}u$ forces all these rotation <derivatives> to vanish, hence $u$ is radial.