Appropriate Boundary Conditions for the Biharmonic Equation

[Arkuski]

Usually, when considering the biharmonic equation (given by $Δ^2u=f$, we look for weak solutions in $H^2_0(U)$, which should obviously have Neumann boundary conditions ($u=0$ and $\bigtriangledown u\cdot\nu =0$ where $\nu$ is normal to $U$).

Now consider that we are looking for solutions $u\in H^1_0(U)\cap H^2(U)$. Clearly, the boundary condition $u=0$ should still apply since we are considering $H^1_0$, but I'm having trouble deciding what the other boundary condition should be. If we multiply the PDE by a test function $v\in H^1_0(U)\cap H^2(U)$ and integrate by parts, we find the following:

$\int _U (Δ^2u)v dx=-(Dv)(Δu)_{∂U}|+\int _U (Δv)(Δu)dx$

If we want the boundary term to vanish, we would require that something like $(\bigtriangledown u\cdot\nu )(Δu)=0$ on the boundary, but I'm confused as to why we require these terms to vanish, or if this is even a necessity.

 

[gtoguy97]

Can someone explain why this boundary condition is necessary, or if there is a different one that should be used instead?