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Let $\Omega \subset \mathbb{R}^2$ be a bounded open domain. Derive the Euler-Lagrange equations and the natural boundary conditions for the two-phase piecewise $H^1$ Mumford- Shah model:

$J(\varphi, u_+, u_-) = \int_{\varphi>0}|f(x) − u_+(x)|^2dx + \int_{\varphi<0}|f(x) − u_-(x)|^2dx + \mu\int_{\Omega}|\nabla H(ϕ(x))|dx + \beta\left(\int_{\varphi>0}|\nabla u_+(x)|^2dx + \int_{\varphi<0}|\nabla u_-(x)|^2dx\right).$

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

How do I work with this problem?

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