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 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.

本文的中心对象是Helfrich函数（的一个特殊版本），它是Willmore函数和面积函数的总和乘以权重因子\（\ varepsilon \ ge 0 \）。我们收集了一些有关Helfrich旋转表面Dirichlet边值问题解的存在性的结果，并涵盖了一些特殊的边界条件和权重因子\（\ varepsilon \ ge 0 \）。这些结果是通过不同的技术获得的，例如能量方法，胶合技术以及在接近Helfrich圆柱体的隐函数定理的使用。特别是，关于边界值的机制，其中，悬链曲面形式存在的区域的全局minimiser功能，该功能赫尔弗里希的minimisers的存在被建立用于所有的权重因子\（\ varepsilon \ GE 0 \） 。对于权重因子\（\ varepsilon \ nearrow \ infty \）的奇异极限，它们均匀地会聚到悬链线上，从而最大程度地减小了旋转表面类别中的表面积。