Nonlinear Schroedinger systems with saturation effect and Willmore boundary value problem

非线性薛定谔系统通常用于描述电磁辐射在由所谓的非线性材料制成的光波导中的传播。当研究克尔介质时，人们通常考虑三次非线性薛定谔系统，在过去的十年中，对该研究领域有许多贡献。在我的研究项目的第一部分，我计划遵循最近由Maia， Montefusco, Pellacci（2013）发表的新想法，他们首先系统地分析了具有饱和效应的Kerr介质模型。我的目的是，首先，锐化它们在这种材料中的非平凡驻波的存在性结果，其次，处理一类更广泛的非线性薛定谔系统，它们同样描述饱和非线性材料。在我的研究项目的第二部分，我计划处理最小弯曲能的曲线和曲面。在数学上，弯曲能量是由威尔默能量量化的，它代表了赫尔弗里希能量的简化变体。在细胞生物学中，最小威莫能量曲线作为细胞膜的模型。近五年来，在满足相同边界条件的所有图形曲线中发现了Willmore能量最小的对称图形曲线。在我的项目中，我希望证明在满足相同边界条件的所有曲线中存在具有最优Willmore能量的曲线。此外，我的目标是将一些已知的关于对称旋转曲面的结果推广到非对称情况和更一般形状的曲面。

Nonlinear Schroedinger systems are commonly used to describe the propagataion of electromagnetic radiation in optic wave guides which are built from so-called nonlinear materials. When Kerr media are investigated one usually considers cubic nonlinear Schroedinger systems and during the past ten years there have been many contributions to that research field. In the first part of my research project I plan to follow new ideas that have been recently published by Maia, Montefusco, Pellacci (2013) who were first to systematically analyze a model for Kerr media with saturation effect. My aim is, firstly, to sharpen their existence results for nontrivial standing waves in such materials and secondly, to deal with a broader class of nonlinear Schroedinger systems that equally describe saturated nonlinear materials.In the second part of my research project I plan to deal with curves and surfaces of minimal bending energy. Mathematically the bending energy is quantified by the Willmore energy which represents a simplified variant of the Helfrich energy. In cell biology curves of minimal Willmore energy serve as a model for cell membranes. In the past five years symmetric graph-shaped curves with minimal Willmore energy among all graph-shaped curves satisfying the same boundary conditions have been found. In my project I wish to prove the existence of curves with optimal Willmore energy among all curves which satisfy the same boundary conditions. In addition I aim at extending some known results for symmetric surfaces of revolution to the nonsymmetric case and to surfaces of a more general shape.