Second variation of the energy functional for adhering vesicles in two space dimensions

Computing the second variation of the energy functional of a lipid vesicle is an issue that precedes any local stability analysis. This issue becomes technically more delicate when the equilibrium configurations to be perturbed are partially adhered to a fixed substrate. Within a two-dimensional model, we introduce here a method that delivers the second variation of the energy functional for vesicles subject to adhesion. The formula thus obtained is then applied to estimate the maximum susceptibility to detachment of tubules, hollow vesicles that do not enclose a prescribed volume. In particular, we illuminate by example the role of the substrate curvature in enhancing or depressing the susceptibility of adhered tubules to a special class of detaching fluctuations.