identically (for $ a \leq -c^2/4$) and the regime where this distribution is non-zero (for a finite distance from the boundary between dense and dilute lipid regions). Fig. 2 summarizes these borderlines in a phase diagram, which is in fact valid also for $b \neq 0$, as will be discussed in the next subsection. The localized regime is shaded in gray and ends at a special point $S$, at which the maximum of the protein distribution is at infinity; as pointed out before, there is an overall symmetry around the $a-$axis ($c=0$).

The effective correlation length $\xi_{eff}$ for the proteins in the localized regime can be estimated from the curvature of $\Phi^0(u)$ at the maximum $u_{max}$, $$\xi_{eff}^{-2} \equiv -h''(u_{max})=2(1-c^2/4)^2$$ This length diverges as one approaches the special point $S$, where the distribution becomes indefinitely broad.

Solution for $b >0\;\;\;\;
-\;\;\;\;$ general considerations
---------------------------------

On physical grounds, the solution for non-zero $b$, i.e., for a finite stiffness of the protein distribution, has to coincide with the solution found for $b=0$ in the preceding section very far from the interface located at $u=0$. This leads to the general boundary condition $$\Phi(u)=\Phi^0(u) \;\;\; \mbox{for} \;\;\;
u \rightarrow \pm \infty$$ where $\Phi^0(u)$ is given 