Where can I find the full derivation of Helfrich's shape equation for closed membranes? I have approximately 10 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/abstract/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.
Edit: So, after some careful thought and hours and hours of work and learning, I realized that the answer that got the bounty was wrong. The author stopped replying to my messages after I gave him bounty.... thanks guys. That said, I've almost got it all figured out (in intense detail) and will post a pdf of my own notes once I'm done!
 A: Edit: Note that I am doing only the first variation, and I am not doing each and every step, mainly those pertinent in understanding how the general shape equation is determined. If you want to see the full derivation, you will need to understand the Geometric Mathematic Primer discussed in Sections 2 and 3 of the book.

*

*Geometric Methods in the Elastic Theory of Membranes in Liquid
Crystal Phases by Zhong-Can Ou-Yang, Ji-Xing Liu, Yu-Zhang Xie, Xie
Yu-Zhang

$c_{0}$: Spontaneous curvature of the membrane surface
$k_{c}$: Bending rigidity of the vesicle membrane
$H$: Mean curvature of the membrane surface at any point $P$
$K$: Gaussian curvature of the membrane surface at any point $P$
$dA$: Area element of the membrane
$dV$: Volume element enclosed by the closed bilayer
$\lambda$: Surface tension of the bilayer, or the tensile strength acting on the membrane
$\Delta p$: Pressure difference between the inside and outside of the membrane.
The shape energy of a vesicle is given by:
$$
F = F_{c} + \Delta p \int dV + \lambda \int dA
$$
Where
$$
F_{c}=\frac{k_{c}}{2}(2H-c_{0})^{2} = \frac{k_{c}}{2}(c_{1}+c_{2}-c_{0})^{2}
$$
The variation of $dA$ and $dV$ are needed, refer to the book to locate those.
Next we'll calculate the first variation of $F$. And we can break this into components by starting with the first variation $F_{c}$.
$$
\delta ^{(1)}F_{c} = \frac{k_{c}}{2}\oint (2H+c_{0})^{2} \delta ^{(1)}(dA) + \frac{k_{c}}{2}\oint 4(2H+c_{0})^{2}(\delta ^{(1)}H)dA
$$
Where the first order variation of $\psi$ gives us:
$$
\delta ^{(1)}dA = -2H\psi g^{1/2}dudv
$$
$$
\delta ^{(1)}dV = \psi g^{1/2}dudv
$$
$$
\delta ^{(1)}H = (2H^{2}-K))\psi + (1/2)g^{ij}(\psi _{ij}-\Gamma _{ij}^{k}\psi_{k})
$$
Note: $\Gamma_{ij}^{k}$ is the Christoffel symbol defined by (for reference):
$$
\Gamma_{ij}^{k} = \frac{1}{2}g^{kl}(g_{il,j} + g_{jl,i} - g_{ij,l})
$$
And we plug those into the variation of $F_{c}$:
$$
\delta ^{(1)}F_{c} = k_{c}\oint [(2H+c_{0})^{2}((2H^{2}-K)\psi + (1/2)g^{ij}(\psi_{ij}-\Gamma_{ij}^{k}\psi_{k}))]
$$
$$
= k_{c}\oint [(2H+c_{0})(2H^{2}-c_{0}H-2K)\psi + (1/2)g^{ij}(2H+c_{0})\psi_{ij} - g^{ij}\Gamma_{ij}^{k}(2H+c_{0})\psi_{k}]g^{1/2}dudv
$$
And there are two relations ($i,j = u,v$)
$$
\oint f\phi_{i}dudv = -\oint f_{i}\phi dudv
$$
$$
\oint f\phi_{ij}dudv = \oint f_{ij}\phi dudv
$$
So then we have:
$$
\delta ^{(1)}F_{c} = k_{c}\oint \left \{ (2H+c_{0})(2H^{2}-c_{0}H-2K)g^{1/2} + [g^{1/2}g^{ij}(2H+c_{0})]_{ij} + [g^{1/2}g^{ij}(2H+c_{0})\Gamma_{ij}^{k}]_{k} \right \}\psi dudv
$$
And we can re-write:
$$
[g^{1/2}g^{ij}(2H+c_{0})]_{ij} = [(g^{1/2}g^{ij})_{j}(2H+c_{0})]_{i} + [g^{1/2}g^{ij}(2H+c_{0})\Gamma_{ij}^{k}]_{k} \psi dudv
$$
And for functions $f(u,v)$, where $u,v = i,j$, we can directly expand:
$$
[(g^{1/2}g^{ij})_{j}f]_{i} = -(\Gamma_{ij}^{k}g^{1/2}g^{ij}f)_{k}
$$
A Laplacian operator for these surfaces is defined in the book, and is given as:
$$
\nabla^{2} = g^{1/2}\frac{\partial }{\partial i}(g^{1/2}g^{ij}\frac{\partial }{\partial j})
$$
So then we have:
$$
[g^{1/2}g^{ij}(2H+c_{0})_{j}]_{i} = g^{1/2}\nabla^{2}(2H+c_{0})
$$
Using these methods in the first variation:
$$
\delta^{(1)}F_{c} = k_{c}\oint [(2H+c_{0})(2H^{2}-c_{0}H-2K) + \nabla^{2}(2H+c_{0})]\psi g^{1/2}dudv
$$
And now we want the variation of $F$.
$$
\delta^{(1)}F = \delta^{(1)}F_{c} + \delta^{(1)}(\Delta p\int dV) + \delta^{(1)}(\lambda\int dA)
$$
Which gives us:
$$
\delta^{(1)}F = \oint [\Delta p-2\lambda H + k_{c}(2H+c_{0})(2H^{2}-c_{0}-2K) + k_{c}\nabla^{2}(2H+c_{0})]\psi g^{1/2}dudv
$$
And since $\psi$ is a very small, well smooth function of $u$ and $v$, the vanishing of the first variation of $F$ requires that:
$$
\Delta p - 2\lambda H + k_{c}(2H+c_{0})(2H^{2}-c_{0}H-2K) + k_{c}\nabla^{2}(2H+c_{0}) = 0
$$
Which is the general shape equation of the vesicle membrane. $c_{0}$ is a constant unless the symmetry effect of the membrane and its environment varies between each point (we assume it doesn't) otherwise $c_{0}$ becomes a function of $u$ and $v$. So we can reduce to:
$$
\Delta p - 2\lambda H + k_{c}(2H+c_{0})(2H^{2}-c_{0}H-2K) + 2k_{c}\nabla^{2}H = 0
$$
Hope this helps. Again I would locate that book to see the full derivations. I don't know if the visible section of the book on Google shows you everything that you need to know, but I surely hope this points you in the right direction to understanding the problem.
A: I studied this problem long time ago (almost ten years) during a training period. It's a really interesting problem which, as far I remember, is not so complicated to understand. It's a basic variational method applied to the curvature. In your notation, $H$ and $K$ are the mean and Gaussian curvature, the two invariants on a (curved) surface. 
Perhaps a good knowledge of Stokes theorem (how to translate volume to surface) and variational calculation is a first step in the understanding. Tensor notation in the theory of surface is also a prerequisite. I was particularly enjoying a book by Do Carmo Differential Geometry of Curves and Surfaces Pearson (1973) at the time of this training period, even if I would no more consider it as a modern text-book.
The strategy is to express the curvatures $c_1$ and $c_2$ in your notation as some differential elements of the metric tensor onto which you can apply usual variation calculus. I think you should have something like $c_{1}+c_{2}=2H$, the mean curvature, since I believe $c_{1,2}$ are the curvatures along the principal axis of the surface, am I correct ? You also need to express the change in the infinitesimal volume and the infinitesimal area in some differential formulas (in term of the metric tensor) that you can vary. Then you should be full of Christophell, and you should try to recognise the Gaussian and mean curvatures of the final result. They may be simplifying physical assumption, like conservation of the volume, which enters the game, too. 
An other way is to express everything in term of the normal vector to the surface. Then, you express the curvature in term of the fundamental form of first and second kind, and make the variational calculation with them. This second method is equivalent to the first one of course, since the fundamental forms can be given in term of the metric. The paper you discuss uses the metric approach if I remember correctly.
I may eventually give you my notes of this period. The only embarrassing point is that they are in French, and I'm really lazy to translate in English :-( The project was to calculate the Green function of a deformed sphere. So it's simpler than yours as believe.
All the details of the calculation can be found in the book by the author: 


*

*Ou-Yang Zhong-Can, Liu Ji-Xing and Xie Yu-Zhang, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases Wolrd Scientific (1999). http://www.worldscientific.com/worldscibooks/10.1142/3579
with some introductory material about variational calculation and differential geometry of surface (called manifold in modern text-book).
A: The paper you're looking for is probably:
"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 
Phys. Rev. A 39, 5280–5288 (1989) 
http://pra.aps.org/abstract/PRA/v39/i10/p5280_1
Hopefully you have university access - I couldn't find a free copy anywhere.  This is actually a more complicated question than most physics papers like to make it appear!  Good luck with your project.  Membranes are a very cool subject.  
A: In the paper by Lin et al. (2003) Progress in Theoretical Physics they mention in the abstract that they extend the work of Ou-yang and Helfrich by expanding the bending energy to fourth order. That means you should be able to work out the lower order solutions from their paper as well.
