  We study critical surfaces for a surface energy which contains the squared
$L^2$ norm of the difference of the mean curvature $H$ and the spontaneous
curvature $c_o$, coupled to the elastic energy of the boundary curve. We
investigate the existence of equilibria with $H\equiv -c_o$.
  When $c_o \ge 0$ we characterize those cases where the infimum of this energy
is finite for topological annuli and we find the minimizer in the cases that it
exists. Results for topological discs are also given.
