Weak and weak-* convergence

Question

If i have a bounded sequence $\{u_n\}_n \in H^1(\Omega) \cap L^\infty(\Omega)$, then by the weak and weak-* compactness of the spaces, there exists subsequences and functions $u_1 \in H^1(\Omega)$, $u_2 \in L^\infty(\Omega)$ such that $u_n \rightharpoonup u_1$ in $H^1$, and $u_n \rightharpoonup u_2$ in $L^\infty$ weak-*. My question is, under what conditions will I get a limit with the same regularity as the sequence? In other words, when is $u_1 = u_2$?

Answer 1

It is easy to check that $u_n \rightharpoonup u_1$ in $H^1$ implies $u_n \rightharpoonup u_1$ in $L^2$. Similarly, $u_n \stackrel*\rightharpoonup u_2$ in $L^\infty$ implies $u_n \rightharpoonup u_2$ in $L^2$. By uniqueness of the weak limit, you have $u_1 = u_2$.