I am reading proof on the partial regularity of harmonic maps and there is a part where the author considers a sequence with the following terms on the unit ball $B$ in $\mathbb{R}^2.$
- $P_{\delta}\in H^{1}(B)$ such that $P_{\delta}\rightharpoonup P$ weakly in $H^{1}(B)$ as $\delta\to 0.$
- $\Omega_{\delta}\in L^{2}(B)$ such that $\Omega_{\delta}\to \Omega$ strongly in $L^2(B)$ as $\delta\to 0.$
- $\xi\in H^{1}_{0}(B)$ such that $\xi\rightharpoonup \xi$ weakly in $H^{1}_0(B)$ as $\delta\to 0.$
Besides this, by Rellich's compactness theorem we can assume (up to subsequence) that $P_{\delta}\to P$ strongly in $L^2(B)$ and therefore also almost everywhere (up to subsequence).
My question is regarding the type and the space in which the following expression converges, $$A_{\delta} = P^{-1}_{\delta} dP_{\delta} + P^{-1}_{\delta} \Omega_{\delta} P_{\delta}.$$
As far as I understand the first term convergences weakly in $L^1(B)$ and the second term converges strongly in $L^2(B)$ and therefore $A_{\delta}$ converges weakly in $L^1(B).$ Is this reasoning correct?