Mathematical analysis of magnetic domain patterns in thin ferromagnetic films

铁磁材料在数据存储技术中起着关键作用，基于其性质形成复杂的磁化模式。 特别感兴趣的是具有垂直各向异性的超薄铁磁膜。通常实验观察到的结构在这样的薄膜是条纹和气泡域相。虽然这些模式已经在物理文献中的特定的anomaly配置的基础上进行了探索，但仍然缺少一个从潜在的微磁能解释这些模式的综合数学理论。因此，这个项目的目的是促进这样一个理论的发展，并从潜在的能量泛函中推导出这些域模式的结构。一个主要的工具是严格推导相关的渐近模型的框架内的伽玛收敛。特别是，我们将解决以下问题：我们首先调查的域模式的形成下的应用程序的外部字段和导出相关的宏观模型。此外，我们将推导出一个渐近模型有关的极限消失膜厚度和分析基态这个模型。我们也将考虑与一维组态有关的离散晶格能。主要目标是显示周期性，基于反射积极性原则。在项目的另一部分，我们将推导出一个有效的薄膜能量，其中考虑到样品边界的影响。最后，我们将解决进一步的问题，例如，分析较厚的薄膜和Dzyaloshinskiii-Moriya相互作用对域图案formation.The分析的挑战，在这个项目源于非凸性，非局部性和矢量字符的基础微磁能。虽然有一些可用的工具，但解决此类复杂问题的一般理论尚不存在。我们使用变分法和渐近分析领域的不同方法，如Gamma收敛，插值估计和几何测度理论的工具。 从更广泛的角度来看，微磁模型可以被视为其他非凸，非局部结构形成系统的原型模型。特别是，我们相信，在这个项目的过程中开发的工具也将适用于相关的图案形成系统，如超导体I+II型模型，超导晶体模型和弹塑性模型的研究。

Ferromagnetic materials play a key role in data storage technologies, based on their property to form complex magnetization patterns. Of particular interest are ultra-thin ferromagnetic films with perpendicular anisotropy. Commonly experimentally observed structures in such films are stripe and bubble domain phases. While these patterns have been explored on the basis of specific ansatz configurations in the physical literature, a comprehensive mathematical theory which explains these patterns from the underlying micromagnetic energy is still missing. The aim of this project is hence to contribute to the development of such a theory and to derive the structure of these domain patterns from the underlying energy functional. One main tool is the rigorous derivation of relevant asymptotic models in the framework of Gamma-convergence.In particular, we will address the following questions: We first investigate the formation of domain patterns under application of an external field and derive related macroscopic models. Furthermore, we will derive an asymptotic model related to the limit of vanishing film thickness and analyze ground states for this model. We will also consider discrete lattice energies related to one-dimensional configurations. The main goal is to show periodicity, based on reflection positivity principles. In another part of the project, we will derive an effective thin-film energy which takes the effect of the sample boundary into account. Finally, we will address further questions as e.g. the analysis of thicker films and the effect of Dzyaloshinskii-Moriya interaction on the domain pattern formation.The analytical challenge in this project stems from the non-convexity, nonlocality and vectorial character of the underlying micromagnetic energy. While there are some tools available, a general theory to solve such complex problems does not yet exist. We use different methods from the fields of calculus of variations and asymptoptic analysis such as Gamma-convergence, interpolation estimates and tools from geometric measure theory. From a broader perspective, the micromagnetic model can be seen as a prototype model for other non-convex, nonlocal structure forming systems. In particular, we believe that the tools developed in the course of this project will be applicable also in the the study of related pattern forming systems such as e.g. superconductor type I+II models, models for nematic crystals and models from elastoplasticity.