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Membranes are an important subject of study in physical chemistry andbiology. They can be considered as material surfaces with a surface energydepending on the curvature tensor. Usually, mathematical models developed inthe literature consider the dependence of surface energy only on mean curvaturewith an added linear term for Gauss curvature. Therefore, for closed surfacesthe Gauss curvature term can be eliminated because of the Gauss-Bonnet theorem.In [18], the dependence on the mean and Gaussian curvatures was considered instatics. The authors derived the shape equation as well as two scalar boundaryconditions on the contact line. In this paper-thanks to the principle ofvirtual working-the equations of motion and boundary conditions governing thefluid membranes subject to general dynamical bending are derived. We obtain thedynamic 'shape equa-tion' (equation for the membrane surface) and the dynamicconditions on the contact line generalizing the classical Young-Dupr{\'e}condition.

Dynamic boundary conditions for membranes whose surface energy depends on the mean and Gaussian curvatures