Weakly lower semicontinuity involving cosine

Question

I want to stablish the existence of $T$-periodic weak solution to $u''+\sin (u)=h$, where $h$ is continuous and $T$-periodic on $\mathbb{R}$ with zero mean. In order to do that I want to find minimum points of $$ J(u)=\frac{1}{2}\int_0^T u'(t)^2\ dt\ + \int_0^T \cos(u(t))\ dt - \int_0^T h(t)u(t)\ dt\qquad u\in H^1_T(\mathbb{R})$$ I have proved the existence of a bounded minimizing sequence $u_n$, and so $u_n$ (up to a subsequence) weakly converges to some $u\in H^1_T(\mathbb{R})$. If I show that $J$ is w.l.s.c. I finish. The first integral is a equivalent norm to the usual one in $H^1_T(\mathbb{R})$, and thus this first term is w.l.s.c. The last term is also w.l.s.c. in view of the weak convergence definition on Hilbert spaces. However I have problems in proving that $$ \liminf \int_0^T \cos(u_n(t))\ dt \ge \int_0^T \cos(u(t))\ dt $$ Maybe I am missing something really easy here. Anyway any help is welcome. Actually if anybody knows some papers/books to read about sufficient conditions to w.l.s.c. please let me know.

Answer 1

Finally, I found a nice solution for those who are interested.

By the Rellich-Kondrachov theorem, the space $H^1 [0,T]$ is compactly embedded in the space of continuous functions $\mathcal{C}[0,T]$. Consequently, $u_n \rightarrow u$ in $\mathcal{C} [0,T]$, that is $\lbrace u_n \rbrace$ uniformly converges to $u$ in $[0,T]$ therefore, also $\lbrace u_n \rbrace$ pointwise converges to $u$ in $[0,T]$. By an obvious application of the dominated convergence theorem, one has in fact $$ \int_0^T \cos (u_n) \ dt \rightarrow \int_0^T \cos (u) \ dt $$