论文标题
特殊Helfrich革命表面功能的边界价值问题
Boundary value problems for a special Helfrich functional for surfaces of revolution
论文作者
论文摘要
本文的中心对象是（特殊版本的）Helfrich功能，它是Willmore功能的总和，而面积功能时间为重量因子$ \ VAREPSILON \ ge 0 $。我们收集了有关革命赫尔夫里奇（Helfrich）革命表面的Dirichlet边界价值问题的解决方案的几个结果，并涵盖了边界条件和权重因子的某些特定制度$ \ VAREPSILON \ ge 0 $。这些结果借助不同的技术，例如能量方法，胶合技术以及接近Helfrich圆柱体的隐式函数定理的使用。特别是，关于边界值的制度，在\ emph {ash aher}重量因子$ \ varepsilon \ ge 0 $的情况下，建立了helfrich函数的最小化函数的最小值。对于重量因子的奇异极限，$ \ varepsilon \近\ infty $它们均匀地汇聚到catenoid，从而最大程度地减少了革命表面中的表面积。
The central object of this article is (a special version of) the Helfrich functional which is the sum of the Willmore functional and the area functional times a weight factor $\varepsilon\ge 0$. We collect several results concerning the existence of solutions to a Dirichlet boundary value problem for Helfrich surfaces of revolution and cover some specific regimes of boundary conditions and weight factors $\varepsilon\ge 0$. These results are obtained with the help of different techniques like an energy method, gluing techniques and the use of the implicit function theorem close to Helfrich cylinders. In particular, concerning the regime of boundary values, where a catenoid exists as a global minimiser of the area functional, existence of minimisers of the Helfrich functional is established for \emph{all} weight factors $\varepsilon\ge 0$. For the singular limit of weight factors $ \varepsilon \nearrow \infty $ they converge uniformly to the catenoid which minimises the surface area in the class of surfaces of revolution.
