Mathematical analysis of magnetic domain patterns in thin ferromagnetic films

铁磁材料具有形成复杂磁化模式的特性，在数据存储技术中起着关键作用。特别感兴趣的是具有垂直各向异性的超薄铁磁薄膜。通常在实验中观察到这种薄膜的结构是条纹和气泡畴相。虽然这些模式已经在物理文献中基于特定的ansatz构型进行了探索，但从潜在的微磁能解释这些模式的综合数学理论仍然缺失。因此，这个项目的目的是促进这样一个理论的发展，并从潜在的能量函数中推导出这些领域模式的结构。一个主要的工具是在伽玛收敛的框架下严格推导相关的渐近模型。特别地，我们将解决以下问题：我们首先研究外场作用下域模式的形成，并推导出相关的宏观模型。此外，我们将推导出一个与消失膜厚度极限有关的渐近模型，并分析该模型的基态。我们还将考虑与一维构型相关的离散晶格能量。主要目标是显示周期性，基于反射正性原则。在项目的另一部分中，我们将推导出考虑样品边界影响的有效薄膜能量。最后，我们将解决进一步的问题，如厚膜的分析和Dzyaloshinskii-Moriya相互作用对畴图案形成的影响。这个项目的分析挑战源于底层微磁能的非凸性、非局域性和矢量性。虽然有一些可用的工具，但解决这种复杂问题的一般理论还不存在。我们使用了变分学和渐近分析领域的不同方法，如伽马收敛、插值估计和几何测量理论的工具。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过程中开发的工具也将适用于相关模式形成系统的研究，例如超导体I+II型模型，向列晶体模型和弹塑性模型。

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.