Let $\;f_n \to f\;$ weakly in $\;W^{1,2} ( (a,b);\mathbb R^m)\;$ where $\;(a,b)\;$ is an open bounded subset of $\;\mathbb R\;$ and $\;W:\mathbb R^m \rightarrow \mathbb R\;$ a non-negative continuous function.
I'm trying to prove this inequality:
$\; \int_{a}^{b} { \vert \nabla f(x)\vert }^2 + W(f(x))\;dx \le \liminf_{n \to \infty} \int_{a}^{b} {\vert \nabla f_n(x) \vert}^2 + W(f_n(x))\;dx\;$
My approach:
Since $\;f_n \to f\;$ weakly in $\;W^{1,2} ( (a,b);\mathbb R^m)\;$ , it follows that $\;\nabla f_n \to \nabla f\;$ weakly in $\;L^2 ( (a,b);\mathbb R^m)\;$ and then it is known :
$\; \int_{a}^{b} { \vert \nabla f(x)\vert }^2 \le \liminf_{n \to \infty} \int_{a}^{b} {\vert \nabla f_n(x) \vert}^2 \;dx\;$
In order to complete the proof I thought that if I could somehow show $\; \int_{a}^{b} W(f(x)) \le \liminf_{n \to \infty} \int_{a}^{b} W(f_n(x)) \;dx\;$ , then using the fact that $\;\liminf(a_n) + \liminf(b_n)\le \liminf(a_n +b_n)\;$ the inequality would be proven.
However I 'm a bit unsure if the above arguments are valid. How should I proceed in order to prove this inequality? What am I missing?
I would appreciate any help! Thanks in advance!
Hint
The function $W$ has all qualifications for Fatou lemma.