朗道-德热纳模型中平面各向同性-向列相界面的尖锐不稳定性
Sharp instability of planar isotropic--nematic interfaces in the Landau--de Gennes model
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中文总结 AI 辅助
该研究在朗道-德热纳模型中移除额外条件后，证明$-\ rac{3}{2}<L<0$区间内对角平面类约化能量的非负极小值解在一维扰动下不稳定，修正了界面轮廓的归一化不一致问题。
中文摘要 AI 辅助
我们研究了具有各向异性弹性常数$L$的朗道-德热纳模型中一维平面各向同性-向列相界面的稳定性。早期工作仅在额外条件下证明了$L<0$时的不稳定性，我们移除了该条件并证明，在$-\ rac{3}{2}<L<0$的整个区间内，对角平面类中约化能量的任意非负极小值解在一般一维扰动下是不稳定的。该结果表明$L=0$是负$L$侧不稳定性范围的尖锐端点：临界算子在$L=0$处非负，而在约化问题的整个负$L$范围内均存在不稳定性。我们还定义了最优稳定性指数，并修正了先前给出的界面轮廓中存在的二倍归一化不一致问题。
英文摘要
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.
