Stationary surfaces with boundaries

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 involves Lagrangian that symmetric in principal curvatures. The first variation computed, and stress tensor extracted whose divergence quantifies deviation from -criticality. Boundary-value problems are then examined, characterization free-boundary -surfaces rotational symmetry given for scaling-invariant -functionals. In case not scaling-invariant, certain boundary-to-interior consequences discussed. Finally, some applications to conformal Willmore energy p-Willmore presented.