I have to solve the following exercise:

Derive the Euler-Lagrange equations (including boundary conditions) for the following minimisation problem:

$$\min J[u] = \displaystyle\int_\Omega \left( \dfrac{1}{2}(\Delta u)^2 + \dfrac{1}{2}\left|\nabla u\right|^2 + \dfrac{1}{2}u^2 - uf(x,y) \right) \mathrm d \Omega\\u(x,y) = u_o(x,y), \text{and }\dfrac{\partial u}{\partial n} = 0,\text{ in } \partial\Omega$$

where $\Omega$ is a bounded domain in $\mathbb{R}^2$ with boundary $\partial\Omega$, consisting of the non-overlapping segments $\partial\Omega_1$ and $\partial\Omega_2$. Further, $f(x,y)$ is a given function.

I think I have some good idea about how I should solve this problem. During class we were told that the strategy is to differentiate $J[u+tv]$ w.r.t. $t$ at $t =0$, where $u$ is the supposed optimal solution of the minimisation problem. After integration by parts we should then be able to apply DuBois-Raymond's lemma to arrive at a PDE with boundary conditions.

However, when I try to evaluate $\dfrac{\partial J[u+tv]}{\partial t}\big|_{t=0}$ I don't understand how I would need integration by parts.

How should I solve this exercise? More specifically, what would $\dfrac{\partial J[u+tv]}{\partial t}\big|_{t=0}$ look like?


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