We consider an energy functional that arises in micromagnetic and liquid crystal theory on thin films. In particular, our energy comprises a non-convex term that models anti-symmetric exchange as well as an anisotropy term. We devise an algorithm for energy minimization in the continuous case and show weak convergence of a subsequence towards a solution of the corresponding Euler--Lagrange equation. Furthermore, an algorithm for numerical energy minimization is presented. We show empirically that this numerical algorithm converges to the correct solutions for a benchmark problem without the need for user-supplied parameters, and present a rigorous convergence analysis for important special cases.

翻译：我们研究一种在薄膜微磁学和液晶理论中出现的能量泛函。该能量泛函特别包含一个用于建模反对称交换作用的非凸项以及一个各向异性项。我们设计了一种用于连续情形下能量最小化的算法，并证明了子序列弱收敛于相应欧拉-拉格朗日方程的解。此外，本文提出了一种数值能量最小化算法。我们通过实验验证该数值算法能在无需用户提供参数的情况下，对基准问题收敛至正确解，并对若干重要特例给出了严格的收敛性分析。