I want to find the strong form of Euler-Lagrange equation of the following variational problem( $\partial\Omega$ smooth ): $$\min_{u\in H^2(\Omega)} I[u]:=\int_\Omega (\partial_x^2u)^2 + 2(\partial_x\partial_yu)^2+(\partial_y^2u)^2 \,dx\,dy$$
I can get the weak form: $$\forall v\in H^2(\Omega),\int_\Omega \partial_x^2u\ \partial_x^2v + 2\partial_x\partial_y u\ \partial_x\partial_y v + \partial_y^2u\ \partial_y^2 v \,dx\,dy=0,$$ i.e. $$\int_\Omega \nabla^2 u:\nabla^2v\,dx\,dy=0.$$ By integration by parts, $$-\int_\Omega div(\nabla^2 u)\cdot \nabla v\,dx\,dy+\int_{\partial\Omega} (\nabla^2u\nabla v)\cdot n\,d\sigma=0$$ ($n$ is outward unit vector, and we assume $\nabla v$ is a column vector) $$\Rightarrow \int_\Omega \Delta^2 u\ v\,dx\,dy - \int_{\partial\Omega} div(\nabla^2 u)\cdot n\ v\,d\sigma+\int_{\partial\Omega} (\nabla^2u\nabla v)\cdot n\,d\sigma=0.$$
Take $v|_\Omega=0,\nabla v|_\Omega=0$ we get $\Delta^2 u=0$; Similarly we get: $$div(\nabla^2 u)\cdot n=\partial_n(\Delta u)=0\ on\ \partial\Omega.$$ But the last boundary condition is confusing me. What can be implied from "for all $v\in H^2(\Omega),$" $$\int_{\partial\Omega} (\nabla^2u\nabla v)\cdot n\,d\sigma=0?$$ Apparently it shouldn't be "$\nabla^2 u\ n=0$",since it contains 2 equations, and it times $\nabla v$ which is a gradient instead of an arbituary vector field. Can anyone help with the last boundary condition? Thanks!