奥森-弗兰克模型中液晶与电场相互作用的变分原理
Variational principles for the interaction of liquid crystals and electric fields in the Oseen--Frank model
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中文总结 AI 辅助
研究单轴向列型液晶与外部电场相互作用，通过变分原理将问题重述为双极小化问题，用矢量势代替标量静电势，量化静电反作用解耦，证明物理极小值收敛到极限调和映射，利于理论分析与数值模拟。
中文摘要 AI 辅助
我们为单轴向列型液晶在单常数奥森-弗兰克近似下与外部电场相互作用开发了一个严格的变分框架。平衡构型由具有挑战性的极小-极大鞍点结构的非局部非线性能量控制。我们的第一个主要结果将此问题重新表述为纯双极小化问题。利用凸对偶性和霍奇分解，我们用矢量势代替标量静电势，得到具有唯一极小值的封闭形式对偶泛函。这种直接能量极小化原理对理论分析和数值模拟都有利。我们的第二个结果严格量化了小介电各向异性极限下静电反作用的解耦。我们在精确非局部能量与其标准局部近似之间建立了一致的二次能量界，从形式上证明了忽略感应去极化场这一广泛的物理启发式方法的合理性。最后，在严格的强制性条件下，我们结合Γ收敛、一致索伯列夫正则性和微扰强制性转移，证明物理极小值以最优的定量线性速率收敛到极限调和映射。
英文摘要
We develop a rigorous variational framework for uniaxial nematic liquid crystals interacting with an external electric field in the one-constant Oseen--Frank approximation. Equilibrium configurations are governed by a nonlocal, nonlinear energy with a challenging min-max saddle-point structure. Our first main result reformulates this problem as a pure double-minimization problem. Using convex duality and the Hodge decomposition, we replace the scalar electrostatic potential with a vector potential, yielding a closed-form dual functional with a unique minimizer. This direct energy-minimization principle is advantageous for both theoretical analysis and numerical simulation. Our second result rigorously quantifies the decoupling of the electrostatic back-reaction in the limit of small dielectric anisotropy. We establish a uniform quadratic energy bound between the exact nonlocal energy and its standard local approximation, formally justifying the widespread physics heuristic of neglecting induced depolarization fields. Finally, under a strict coercivity condition, we combine $Γ$-convergence, uniform Sobolev regularity, and a perturbative coercivity transfer to prove that physical minimizers converge to limiting harmonic maps at an optimal, quantitative linear rate.
