= Solution
At every point the <deformation gradient> attains the differentiable energy's minimum, so $DW(D\mathbf y)=0$ and it is a natural state. By the strict Baker-Ericksen argument just proved, all its principal stretches are equal. In the radial formula this gives
$$
r'(R)=r(R)/R,\qquad\frac d{dR}\left(\frac{r(R)}R\right)=0.
$$
Thus $r(R)=vR$ for one constant $v>0$ on the <connected> radial interval, and
$$
\boxed{\mathbf y(\mathbf x)=v\mathbf x.}
$$
The constancy is global through the annulus, not a separately chosen dilation at each point.
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