Symmetric Willmore surfaces of revolution satisfying natural boundary conditions

We consider the Willmore-type functional $$\mathcal{W}_{\gamma}(\Gamma):= \int\limits_{\Gamma} H^2 dA -\gamma \int\limits_{\Gamma} K dA,$$ where H and K denote mean and Gaussian curvature of a surface Γ, and ${\gamma \in 0,1]}$ is a real parameter. Using direct methods of the calculus of variations, we prove existence of surfaces of revolution generated by symmetric graphs which are solutions of the Euler-Lagrange equation corresponding to ${\mathcal{W}_{\gamma}}$ and which satisfy the following boundary conditions: the height at the boundary is prescribed, and the second boundary condition is the natural one when considering critical points where only the position at the boundary is fixed. In the particular case γ = 0 these boundary conditions are arbitrary positive height α and zero mean curvature.