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.
