(a) State the Lax-Milgram lemma. Use it to prove that there exists a unique function $u$ in the space

$$H_{\partial}^{2}(\Omega)=\left\{u \in H^{2}(\Omega) ;\left.u\right|_{\partial \Omega}=\partial u /\left.\partial \gamma\right|_{\partial \Omega}=0\right\}$$

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary and $\gamma$ its outwards unit normal vector, which is the weak solution of the equations

$$\begin{aligned} \Delta^{2} u &=f \text { in } \Omega, \\ u &=\frac{\partial u}{\partial \gamma}=0 \text { on } \partial \Omega, \end{aligned}$$

for $f \in L^{2}(\Omega), \Delta$ the Laplacian and $\Delta^{2}=\Delta \Delta$.

[Hint: Use regularity of the solution of the Dirichlet problem for the Poisson equation.]

(b) Let $\Omega \subset \mathbb{R}^{n}$ be a bounded domain with smooth boundary. Let $u \in H^{1}(\Omega)$ and denote

$$\bar{u}=\int_{\Omega} u d^{n} x / \int_{\Omega} d^{n} x$$

The following Poincaré-type inequality is known to hold

$$\|u-\bar{u}\|_{L^{2}} \leqslant C\|\nabla u\|_{L^{2}}$$

where $C$ only depends on $\Omega$. Use the Lax-Milgram lemma and this Poincaré-type inequality to prove that the Neumann problem

$$\begin{aligned} &\Delta u=f \text { in } \Omega \\ &\frac{\partial u}{\partial \gamma}=0 \text { on } \partial \Omega \end{aligned}$$

has a unique weak solution in the space

$$H_{-}^{1}(\Omega)=H^{1}(\Omega) \cap\{u: \Omega \rightarrow \mathbb{R} ; \bar{u}=0\}$$

if and only if $\bar{f}=0$.