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.