In the notes I am trying to follow one can find the following argument (part of a longer proof on existence of a weak solution to a certain type of nonlinear elliptic pde):
Let $V = H^1_0(\Omega)$ and consider $(u_m) \subset V$ such that $||u_m||_V \leq C$ for all m. It follows that (after some abuse in labelling) $u_{m_k} \to u$ weakly in V, $u_{m_k} \to u$ in $L^2(\Omega)$ and $u_{m_k} \to u$ a.e. in $\Omega$. Furthermore, let $v \in V$ with $v_m \to v$ in $V$. Finally, $a(x,u)$ is a Caratheodory function (so in particular continuous w.r.t to the second variable) with uniform bounds $A_1 \leq a \leq A_2$. Then, apparently, we use Dominated Convergence Theorem (DCT) to conclude that $$ a(\cdot,u_{m_k})\nabla v_{m_k} \to a(\cdot,u)\nabla v \text{ in } L^2(\Omega), $$ which also apparently allows us to conclude that $$ \int_{\Omega} a(x,u_{m_k})\nabla u_{m_k} \cdot \nabla v_{m_k} \to \int_{\Omega} a(x,u)\nabla u \cdot \nabla v. $$ First of all, I reckon that in the first line there is a typo and we have $u_{m_k}$ instead of $v_{m_k}$, as the way it is I suppose we do not need DCT to show $L^2$ convergence (adding zero cleverly seems to be enough). If there indeed is a typo there, then how can we apply DCT? In its form stated in the notes, we would need a.e. convergence of LHS to RHS but for that we would need a.e. convergence of $\nabla u_{m_k}$ - does it follow from a.e. convergence of $u_{m_k}$? If there is no typo, then how can we go from first line to the second? Many thanks for any insight you might have!
I don't think this is a typo.
Denote the $m_k$ as $k$. $$a(\cdot, u_k) \nabla v_k \rightarrow a(\cdot, u) \nabla v \quad \text{ strongly in } L^2$$ lets call this part $f_k \rightarrow f$ in $L^2$, and $$\nabla u_k \rightharpoonup \nabla u \quad \text{ weakly in } L^2$$ lets call this part $g_k \rightharpoonup g$ in $L^2$. What you want to show in the end is that $\lim_k \bigg|\int f_k g_k - \int f g\bigg|= 0.$ $$\bigg|\int f_k g_k - \int f g\bigg| \\ \leq\bigg|\int f_k g_k - \int fg_k\bigg| + \bigg|\int fg_k - \int f g\bigg|\\\leq \|f_k - f\|_2 \|g_k\|_2 + \int f(g_k - g). $$ Since the sequence $g_k$ weakly converges, its norm $\|g_k\|_2$ is bounded. And we see the last line goes to zero as $k$ goes to infinity.
The argument $u_k \rightarrow u$ a.e. is used in order to show $a(\cdot, u_k) \rightarrow a(\cdot, u)$ a.e. so we can apply DCT to $a(\cdot, u_k) \nabla v_k$.