I came across this problems i few months and go and thought i would give it a try. The problem goes as following:
Let $\Omega$ be a bounded subset of $R^n$, $\alpha>0, f \in L^2(\Omega)$ and we have:
(1) \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 left some images of what I managed to do so far, but i don't understand how i'm suppose to show that $u_\alpha \rightarrow u_D$ in $H^1(\Omega)$ or $u_\alpha \rightarrow u_N$ in $H^1(\Omega)$. I tried to prove that $||u_{\alpha}-u_D||^2_{H^1(\Omega)}\rightarrow 0$ when when $\alpha \rightarrow \infty$ and $||u_{\alpha}-u_N||^2_{H^1(\Omega)}\rightarrow 0$ when $\alpha \rightarrow 0$ but i can't get to this.
Thanks for your help.
