I have a bunch of papers that claim that, from the equation for shape energy:
$$ F = \frac{1}{2}k\_c \int (c\_1+c\_2-c\_0)^2 dA + \Delta p \int dV + \lambda \int dA$$
one can use "methods of variational calculus" to derive the following:
$$\Delta p - 2\lambda H + k(2H+c\_0)(2H^2-2K-c\_0H)+2k\nabla^2H=0$$
But I'm having a lot of trouble tracking down the original derivation. The guy who did it first was Helfrich, and here's his and Ou-yang's paper deriving it:
<http://prl.aps.org/pdf/PRL/v59/i21/p2486_1> . However, they don't show an actual derivation, instead saying "the derivation will appear in a full paper by the authors" or something like that. Yet everybody cites the paper I just linked for a derivation. Does anybody know a source that can derive this, or can give me some hints to figure it out myself? To be honest I can't even figure out how to find the first variation. Thanks!



Deriving Helfrich's shape equation for closed membranes
The follow-up paper by Helfrich you are searching for is 


*Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders,*
Ou-Yang Zhong-can and Wolfgang Helfrich,
Physical Review A **39**, 5280-5288 (1989). 



> 
> A general equation of mechanical equilibrium of fluid membranes
>  subject to bending elasticity [reported in Phys. Rev. Lett. 59, 2486
>  (1987)] is derived in detail.
> 
> 
> 


