Differentiating equations for given model

[Railgun]

I need to derive Euler-Lagrange equations and natural boundary conditions for a given model. I've worked out and broken down the model into the following 5 parts:

 

J1 = ∫ {ϕ>0} |f(x) − u+(x)|^2dx

 

J2 = ∫ {ϕ<0} |f(x) − u-(x)|^2dx

 

J3 = ∫ Ω |∇H(ϕ(x))|dx

 

J4 = ∫ {ϕ>0} |∇u+(x)|^2dx

 

J5 = ∫ {ϕ<0} |∇u-(x)|^2dx.

 

where f : Ω → R and u+- ∈ H^1(Ω) (functions such that ∫ Ω(|u|^2 + |∇u|^2)dx < ∞).

 

I need to differentiate each of these 5 equations in terms of ϕ,u+ and u-, any assistance would be very appreciated as I'm weak in calculus.

 

 

I tried getting the first variation, for example for J1(ϕ),

 

let v be a perturbation defined in a space V such that J(v) exists.

 

deltaJ1(ϕ) = lim{epilson->0} [J1(ϕ+epilson.v)-J1(ϕ)]/epilson

 

= lim{epilson->0} ∫{ϕ+epilson.v>0}lf(x)-u+(x)l^2dx-∫{ϕ>0}lf(x)-u+(x)l^2dx From here onwards I'm not sure how to proceed.