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Differentiating equations for given model

  1. Mar 6, 2012 #1
    1. The problem statement, all variables and given/known data

    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.


    2. Relevant equations



    3. The attempt at a solution

    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.
     
  2. jcsd
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