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The central object of this article is (a special version of) the Helfrichfunctional which is the sum of the Willmore functional and the area functionaltimes a weight factor $\varepsilon\ge 0$. We collect several results concerningthe existence of solutions to a Dirichlet boundary value problem for Helfrichsurfaces of revolution and cover some specific regimes of boundary conditionsand weight factors $\varepsilon\ge 0$. These results are obtained with the helpof different techniques like an energy method, gluing techniques and the use ofthe implicit function theorem close to Helfrich cylinders. In particular,concerning the regime of boundary values, where a catenoid exists as a globalminimiser of the area functional, existence of minimisers of the Helfrichfunctional is established for \emph{all} weight factors $\varepsilon\ge 0$. Forthe singular limit of weight factors $ \varepsilon \nearrow \infty $ theyconverge uniformly to the catenoid which minimises the surface area in theclass of surfaces of revolution.

Boundary value problems for a special Helfrich functional for surfaces of revolution