Let $u$ a harmonic function in $B_{1}(0)=\{x \in \mathbb{R}^{n}; |x| <1\}$, with $u(x)=v(r)$ in $\partial U$ then $u(x)=v(r)$, $\forall x \in B_1{(0)}$, i.e, a harmonic function radial on unit sphere is radial in its interior.
My idea was to use the principle of maximum, but i imagine it's more than that.
Thanks!