  Lipid domains are found in systems such as multi-component bilayer membranes
and single component monolayers at the air-water interface. It was shown by
Andelman et al. (Comptes Rendus 301, 675 (1985)) and McConnell et al. (Phys.
Chem. {\bf 91}, 6417 (1987)) that in monolayers, the size of the domains
results from balancing the line tension, which favors the formation of a large
single circular domain, against the electrostatic cost of assembling the
dipolar moments of the lipids. In this paper, we present an exact analytical
expression for the electric potential, ion distribution and electrostatic free
energy for different problems consisting of three different slabs with
different dielectric constants and Debye lengths, with a circular homogeneous
dipolar density in the middle slab. From these solutions, we extend the
calculation of domain sizes for monolayers to include the effects of finite
ionic strength, dielectric discontinuities (or image charges) and the
polarizability of the dipoles and further generalize the calculations to
account for domains in lipid bilayers. In monolayers, the size of the domains
is dependent on the different dielectric constants but independent of ionic
strength. In asymmetric bilayers, where the inner and outer leaflets have
different dipolar densities, domains show a strong size dependence with ionic
strength, with molecular-sized domains that grow to macroscopic phase
separation with increasing ionic strength. We discuss the implications of the
results for experiments and briefly consider their relation to other two
dimensional systems such as Wigner crystals or heteroepitaxial growth.
