= Radial zero-energy rigidity
{title2=$W(Dy)=0\implies y(x)=vx$}
Consider an orientation-preserving classical <radial deformation of an elastic solid> on a connected spherical annulus. If its nonnegative energy is zero everywhere and the material is isotropic with strict <Baker-Ericksen inequalities>, every gradient is a natural state, so $r'=r/R$. Thus $(r/R)'=0$ and $y=vx$.
A separate rigidity argument applies when the zero wells are $SO(3)\cup\bigcup_{i=1}^NSO(3)U_i$, with finitely many positive definite symmetric $U_i$. The radial gradient $F$ is positive symmetric; if $F=QU_i$, then $F^2=U_i^2$, so uniqueness of the positive square root gives $F=U_i$. The rotation well similarly gives $F=I$. Hence the continuous ratio $r(R)/R$ takes values in a finite set of eigenvalues and must be constant on the connected radial interval. Differentiating $r=vR$ finishes the proof. This version needs neither material isotropy nor Baker-Ericksen inequalities and does not require an assumed continuity of $r'$ beyond the radial-map regularity used for the ratio.
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