Kidney-boojum-like solutions and exact shape equation of lipid monolayer domains

The shape of the solid lipid monolayer domain surrounded by a fluid phase is of considerable interest from both physical and mathematical points of view. Here, we report two new results about this topic. First, we find an exact analytical solution to an approximated shape equation that we derived recently. This solution can well-describe the kidney- and boojum-like domains that abound in lipid monolayer. Second, we derive an exact domain shape equation by a direct variation of domain energy without any artificial cutoff. We find no continuous solutions that satisfy this shape equation due to the divergence of its coefficients, which is rooted in the continuous description of electrostatic dipoles.