Within the framework of the generalised Landau-de Gennes theory, we identify a Q-tensor-based energy that reduces to the four-constant Oseen–Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the Q-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimisation problem is well-posed for a significantly wider choice of elastic constants. In particular, this quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen–Frank counterpart via a Г-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimisers.

中文翻译：

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具有四次弹性项的新型 Landau-de Gennes 模型

在广义 Landau-de Gennes 理论的框架内，我们确定了一个问-基于张量的能量，当它被认为是可定向的单轴向列态时，它减少到四常数 Oseen-Frank 能量。虽然通常认为的朗道-德热讷理论版本的弹性贡献在问-张量及其导数，这里提供的替代方案是这些变量的四次。与三次理论相比，我们的方法的一个明显优势是相关的最小化问题非常适合弹性常数的更广泛选择。特别是，这种四次能量可用于模拟高度不同的弹性常数的向列相到各向同性相变。除了证明所提出的 Landau-de Gennes 理论版本的适定性之外，我们还通过在消失向列相关长度的限制中的 Г-收敛论证，在该理论与其 Oseen-Frank 对应理论之间建立了严格的联系。我们还证明了相关最小化器的强收敛性。