Sharp instability of planar isotropic--nematic interfaces in the Landau--de Gennes model

We study the stability of one-dimensional planar isotropic–nematic interfaces in the Landau–de Gennes model with anisotropic elastic constant $L$. Earlier work proved the instability for $L<0$ only under an extra condition. We remove this condition and prove that any non-negative minimizer of the reduced energy within the diagonal planar class is unstable under general one-dimensional perturbations throughout $-\frac{3}{2}<L<0$. This result shows that $L=0$ is the sharp endpoint of the instability range from the negative-$L$ side: the critical operator is non-negative at $L=0$, while instability holds over the full negative-$L$ range of the reduced problem. We also define the optimal stability index and correct a factor-of-two normalization inconsistency in a previously stated interface profile.