Effects of surface tension and elasticity on critical points of the Kirchhoff–Plateau problem

We introduce a modified Kirchhoff–Plateau problem adding an energy term to penalize shape modifications of the cross-sections appended to the elastic midline. In a specific setting, we characterize quantitatively some properties of minimizers. Indeed, choosing three different geometrical shapes for the cross-section, we derive Euler–Lagrange equations for a planar version of the Kirchhoff–Plateau problem. We show that in the physical range of the parameters, there exists a unique critical point satisfying the imposed constraints. Finally, we analyze the effects of the surface tension on the shape of the cross-sections at the equilibrium