  Shapes of planar lipid monolayer domains at the air-water interface are
theoretically and numerically investigated by minimizing the formation energy
of the domains which consist of the surface energy, line tension energy, and
dipole electrostatic energy. The shape equation which describes boundary curves
of the domains at equilibrium state is derived from the first order variation
of the formation energy. A relaxation method is proposed to find the numerical
solutions of the shape equation. The theoretical and numerical results are in
good agreement with previous experimental observation. Some new shapes not
observed in previous experiments are also obtained, which awaits experimental
confirmation in the future.
