The hedgehog uniquely minimises every $p$-energy

We prove that for every dimension $n\ge 2$ and parameter $p\in ]1,n[$, the hedgehog $u_0(x)=x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ is the unique minimiser of the $p$-energy ${\mathcal{E}}_p[u]:={\int }_{{\mathbf{B}}^n}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p\mathrm{d}x$ in the class of $W^{1,p}$-mappings from the $n$-dimensional unit ball ${\mathbf{B}}^n$ to the $(n-1)$-dimensional unit sphere ${\mathbf{S}}^{n-1}=\mathrm{\partial }{\mathbf{B}}^n$ whose trace equal to the identity on ${\mathbf{S}}^{n-1}$. This answers in the affirmative an open question on p.29 in [Robert M. Hardt, Singularities of harmonic maps, Bull. Amer. Math. Soc. (N.S.) 34 (1997), 15–34].