= Solution
The fixed-displacement <boundary value problem> now has $u=0$ on the whole boundary. The <elastic normal-mode energy identity> still holds, because its boundary term contains $\overline u$. Define
$$
B_{\alpha i\beta j}=K_{k\gamma}\epsilon_{\alpha\beta\gamma}\epsilon_{ijk},\qquad H_{\alpha i\beta j}=c_{\alpha i\beta j}+B_{\alpha i\beta j}.
$$
For real constant $K$, $B$ has major symmetry: both <Levi-Civita symbols> change sign upon exchanging the two index pairs. Its quadratic density is a <quadratic minor null Lagrangian>. Indeed,
$$
\begin{aligned}
\int_{\Omega_0}\overline{u_{i,\alpha}}B_{\alpha i\beta j}u_{j,\beta}\,dV
&=\int_{\partial\Omega_0}\nu_\alpha\overline{u_i}B_{\alpha i\beta j}u_{j,\beta}\,dS\\
&\quad-\int_{\Omega_0}\overline{u_i}K_{k\gamma}\epsilon_{\alpha\beta\gamma}\epsilon_{ijk}u_{j,\beta\alpha}\,dV=0.
\end{aligned}
$$
The boundary integral vanishes by the <Dirichlet boundary condition>, and the volume integral vanishes because mixed derivatives are symmetric in $\alpha,\beta$ whereas $\epsilon_{\alpha\beta\gamma}$ is antisymmetric. Constancy of $K$ is essential: otherwise its derivatives would produce extra terms. This calculation initially applies to smooth fields and extends to fields with zero Sobolev boundary trace by density.
It follows that the <Rayleigh quotient> numerator can be replaced by the quadratic form with $H$:
$$
\omega^2\int_{\Omega_0}\rho_0|u|^2\,dV=\int_{\Omega_0}\overline{u_{i,\alpha}}H_{\alpha i\beta j}u_{j,\beta}\,dV.
$$
The assumed positivity of $H$ on all nonzero real matrices also gives positivity on nonzero complex matrices by splitting real and imaginary parts. A nonzero $u$ satisfying $u=0$ on the boundary cannot have identically zero gradient. Hence the right side is strictly positive, and
$$
\boxed{\omega^2>0\quad\text{for every nonzero fixed-boundary mode}.}
$$
This works even when the original moduli quadratic form is not pointwise positive on all matrices: its integrated energy on fixed-boundary gradients agrees with a positive density after adding the <quadratic minor null Lagrangian>.
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