in the aqueous subphase, the protein chemical potential $\mu$ corresponds to the free energy of adsorbing proteins from the subphase into the monolayer and depends on the concentration of proteins in the subphase; this is discussed in Sect. VIII. Since we consider an insoluble (Langmuir) monolayer, similar considerations do not apply to the chemical potential $\mu_{\eta}$ of the lipid order parameter $\eta$. In fact, $\mu_{\eta}$ will be uniquely determined by the requirement of coexistence between dense and dilute lipid regions. For proteins which are [*insoluble*]{} in the subphase, the chemical potential $\mu$ acts as a Lagrange multiplier fixing the total amount of protein in the monolayer, which is a conserved quantity in this situation.

In the LE/LC two-phase region, obtained for $J>0$, one finds experimentally[@Mohwald] domains of typically circular shape of LC phase immersed in a background of LE phase. Since the domains are rather large ($\sim 10-100 \mu m$), we neglect the shape of the line boundary between the LC and LE regions and assume variation of the lipid concentration only along one spatial direction (the $x$ direction) and translational invariance along the perpendicular direction. The free energy $\gamma$ per unit length of this line boundary (related to the 