I recently was given an exercise to complete along the lines of:
Given $u_{\lambda}$ is a weak solution of,
\begin{align}
-\Delta u_{\lambda}+\lambda mu_{\lambda}&=f\in L^{2}(\Omega)\quad f\geq 0,\\
u&=0\text{ on }\partial\Omega,
\end{align}
with $m\in L^{\infty}(\Omega)$, $\Omega\subseteq\mathbb{R}^{N}$ bounded and smooth and $\lambda\in\mathbb{R}$, $\lambda>0$. Show that $u_{\lambda}\rightharpoonup u_{\infty}$ in $H^{1}(\Omega)$.
My question is, can you ever expect that the weak limit of a sequence of solutions preserves the boundary conditions of weak solutions? That is, are there any conditions such that $u_{\infty}=u_{\lambda}$ on $\partial\Omega$ for all $\lambda$?
Best Answer
The set $\{u \in H^1(\Omega) | u(x)=0 \; x \in \partial \Omega \}$ is closed and convex. For this purpose, note that that the set is closed as the pre-image of $0$ under the continuous trace operator. It is also convex. By Mazurs Theorem, it is weakly closed.