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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:

$$J_1 = \int_{\varphi>0}|f(x) − u_+(x)|^2dx$$

$$J_2 = \int_{\varphi<0}|f(x) − u_-(x)|^2dx$$

$$J_3 = \int_{\Omega}|\nabla H(\varphi(x))|dx$$

$$J_4 = \int_{\varphi>0}|∇u_+(x)|^2dx$$

$$J_5 = \int_{\varphi<0}|∇u_-(x)|^2dx$$

where $f : \Omega \to \mathbb{R}$ and $u_+$, $u_- \in H^1(\Omega)$ (functions $u$ such that $\int_{\Omega}(|u|^2 + |\nabla u|^2)dx < \infty$).

I need to differentiate each of these $5$ equations in terms of $\varphi$, $u_+$ and $u_-$, any assistance would be very appreciated as I'm weak in calculus. The idea suggested to me was to get the first variation, for example for $J_1(\varphi)$.

Let $v$ be a perturbation defined in a space $V$ such that $J(v)$ exists, then

$$\delta J_1(\varphi) = \lim_{\varepsilon\to 0} \frac{J_1(\varphi + \varepsilon\cdot v)-J_1(\varphi)}{\varepsilon} = \lim_{\varepsilon\to 0}\frac{1}{\varepsilon}\left(\int_{\varphi+\varepsilon\cdot v>0}|f(x) - u_+(x)|^2 - \int_{\varphi>0}|f(x) - u_+(x)|^2\right).$$

From here onwards I'm not sure how to proceed.

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You should write in Tex –  Riccardo.Alestra Mar 6 '12 at 9:48
    
Do you mean LaTex or Text? –  Shinpachi Mar 6 '12 at 16:20

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