This is a homework question so giving the full answer is not the intention. Rather, I am looking for a hint.
I am asked to minimise the functional $F[u] = \int_\Omega\: \frac{1}{2} E(x,y) \left( \Delta u\right)^2 + q(x,y) u \: d\Omega$. Here, $\Delta u = \nabla \cdot \nabla u$. $\Gamma$ is the boundary of $\Omega$, $\Gamma = \Gamma_1 \cup \Gamma_2$.The boundary conditions are that $\nabla u \cdot \mathbf{n} = 0$ on $\Gamma_1$ and $u = g(x,y)$ on $\Gamma_1$.
Previously, I've used the Gâteaux derivative $dF[u;\eta]$, where of course $\eta = 0$ on $\Gamma_1$ so that the variation around the solution satisfies the boundary condition.
Anyway, I get as a result that $dF[u;\eta] = \int_\Omega\: E(x,y) \Delta u \Delta \eta + q(x,y)\eta \: d\Omega$. And that's where I'm stuck; In the previous problems I would've searched for a term $\nabla \cdot \mathbf{z}$ so that I could replace the integrand by that divergence minus some missing term. This would then lead, after applying Guass's theorem, to some integral over the boundary equals an integral over the domain; we set the boundary integral to zero and using the (extended?) Dubois-Reymond lemma we then find the euler-lagrange equation, where the boundary integral gives us the natural boundary conditions.
Sadly, I can find no such term that is sufficient to remove derivatives of $\eta$ from the Gâteaux derivative. I am looking for a hint so that I can derive the euler-lagrange equations and the natural boundary conditions.
Thanks in advance, Daimonie
PS: The extended dubois-reymond lemma is not provided and I cannot find a good statement of it.
In the question, I mentioned that 'of course' $\eta = 0$ where $ u = g$, so that $u = \hat{u} + \epsilon \eta$ fits the BC. Looking at how I phrased this question, I did not see that $\frac{\partial \eta}{\partial n} = 0$ on the other boundary. That removes the $\nabla \eta$ terms, so that it should work.
Mind you, I turned this in weeks ago; I'm posting here so that it's no longer an open question.