Nonlinear Schroedinger systems with saturation effect and Willmore boundary value problem

非线性薛定谔系统通常用来描述电磁辐射在由所谓的非线性材料构成的光波导中的传播。当研究克尔介质时，人们通常会想到立方非线性薛定谔系统，在过去的十年里，人们对这一领域的研究做出了许多贡献。在我的研究项目的第一部分，我计划遵循Maia，Montefusco，Pellacci(2013)最近发表的新想法，他们首先系统地分析了具有饱和效应的Kerr媒体的模型。我的目的是，首先，加强它们在这类材料中非平凡驻波的存在性结果；其次，处理一类更广泛的、等价地描述饱和非线性材料的非线性薛定谔系统。在我的研究项目的第二部分，我计划处理弯曲能最小的曲线和曲面。在数学上，弯曲能由威尔莫尔能量来量化，威尔莫尔能量代表赫尔夫里奇能量的简化变体。在细胞生物学中，最小威尔莫尔能量曲线可作为细胞膜的模型。在过去的五年中，人们在所有满足相同边界条件的图形曲线中找到了具有最小Willmore能量的对称图形曲线。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，我的目的是将对称旋转曲面的一些已知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.