i) State the Lax-Milgram lemma.

ii) Consider the boundary value problem

$$\begin{aligned} \Delta^{2} u-\Delta u+u=f & \text { in } \Omega \\ u=\nabla u \cdot \gamma=0 & \text { on } \partial \Omega \end{aligned}$$

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with a smooth boundary, $\gamma$ is the exterior unit normal vector to $\partial \Omega$, and $f \in L^{2}(\Omega)$. Show (using the Lax-Milgram lemma) that the boundary value problem has a unique weak solution in the space

$$H_{0}^{2}(\Omega):=\{u: \Omega \rightarrow \mathbb{R} ; u=\nabla u \cdot \gamma=0 \text { on } \partial \Omega\}$$

[Hint. Show that

$$\|\Delta u\|_{L^{2}(\Omega)}^{2}=\sum_{i, j=1}^{n}\left\|\frac{\partial^{2} u}{\partial x_{i} \partial x_{j}}\right\|_{L^{2}(\Omega)}^{2} \quad \text { for all } u \in C_{0}^{\infty}(\Omega)$$

and then use the fact that $C_{0}^{\infty}(\Omega)$ is dense in $\left.H_{0}^{2}(\Omega) .\right]$