I want to show that W.L.S.C. (weakly lower semi continuous) implies L.S.C. (lower semi continuous) and
I have a question:
If $f_n\rightarrow f$ weakly, and $f_n\rightarrow f$ strongly, is there any relation between $\liminf f_n$ in these cases, for example $\liminf f_n$ in weakly convergence is smaller than that of the strongly convergence?
Best Answer
It is easy to see, if you write the definitions (statements) of W.L.S.C. and L.S.C. as follows:
W.L.S.C.: If $u_n \rightharpoonup u$ (weakly), then $\phi(u) \leq \liminf_{n \rightarrow \infty} u_n$.
L.S.C.: If $u_n \rightarrow u$ (strongly), then $\phi(u) \leq \liminf_{n \rightarrow \infty} u_n$.
Then the implication W.L.S.C. $\Rightarrow$ L.S.C. is easy to see:
Let $u_n \rightarrow u$ then $u_n \rightharpoonup u$ and using assumption of W.L.S.C. we have $\phi(u) \leq \liminf_{n \rightarrow \infty} u_n $.