I'm trying to solve the following problem which derives from problem 5 in chapter 6 of Evans Book "Partial Different Equations":
Let $\Omega$ be a bounded subset of $R^n$, $\alpha>0, f \in L^2(\Omega)$ and we have:
\begin{cases} -\Delta u_\alpha= f \ \textrm{ in }\ \Omega \\ \alpha u_\alpha + \frac{\partial u_\alpha}{\partial \nu}=0 \ \textrm{ on } \ \partial \Omega\end{cases}
(a) Prove that when $\alpha \rightarrow \infty$ then $u_\alpha \rightarrow u_D$ in $H^1(\Omega)$ where $u_D$ is a solution of:
(D) \begin{cases} -\Delta u= f \ \textrm{ in } \ \Omega \\ u=0 \ \textrm{on} \ \partial \Omega\end{cases}
(b) Prove that when $\alpha \rightarrow 0$ then, if $f \perp1$ (which is the same as $\int_\Omega f=0)$ , we have $u_\alpha \rightarrow u_N$ in $H^1(\Omega)$ where $u_N$ is a solution of:
(N) \begin{cases} -\Delta u= f \ \textrm{ in } \ \Omega \\ \frac{\partial u}{\partial \nu}=0 \ \textrm{ on } \ \partial \Omega\end{cases}.
I manage to solve problem 5 of Evans Book but so far I'm not be able to make progress on this exercise.