I want to know the bi-Laplace equation with this two boundary conditions admits an unique solution in $H^2(\Omega)$, for given $h\in H^{-1/2}(\partial \Omega)$.
$$\Delta ^2 u=0,\quad on\quad \Omega, $$ $$\partial_n (\Delta u)\Big|_{\partial \Omega}=0,\qquad \Delta u\Big|_{\partial \Omega}=h,$$ where $\partial_n$ denotes the normal derivative, $n$ is the outward unit normal to $\partial \Omega$.