  This article investigates stationary surfaces with boundaries, which arise as
the critical points of functionals dependent on curvature. Precisely, a
generalized "bending energy" functional $\mathcal{W}$ is considered which
involves a Lagrangian that is symmetric in the principal curvatures. The first
variation of $\mathcal{W}$ is computed, and a stress tensor is extracted whose
divergence quantifies deviation from $\mathcal{W}$-criticality. Boundary-value
problems are then examined, and a characterization of free-boundary
$\mathcal{W}$-surfaces with rotational symmetry is given for scaling-invariant
$\mathcal{W}$-functionals. In case the functional is not scaling-invariant,
certain boundary-to-interior consequences are discussed. Finally, some
applications to the conformal Willmore energy and the p-Willmore energy of
surfaces are presented.
