  We perform a variational analysis of an elastic membrane spanning a closed
curve which may sustain bending and torsion. First, we deal with parametrized
curves and linear elastic membranes proving the existence of equilibria and
finding first-order necessary conditions for minimizers computing the first
variation. Second, we study a more general case, both for the boundary curve
and for the membrane, using the framed curve approach. The infinite dimensional
version of the Lagrange multipliers' method is applied to get the first-order
necessary conditions. Finally, a numerical approach is presented and employed
in several numerical test cases.
