I want to show that the functional $L(u)=\int_0^1 \sqrt{1+(u'(x))^2} dx$ is lower semicontinuous in terms of weak convergence in $W^{1,p}(0,1), p\in(1,\infty)$ but not continuous. Our definition of weak lower semicontinuous is: A function $F:X \rightarrow \mathbb{R}$ is weak lower semicontinuous, if $u_k \rightarrow u$ weakly in $X$, then $F(u) \leq \liminf\limits_{k \rightarrow \infty} F(u_k)$. Hint: For a counterexample to the weak continuity, try to approximate a constant function with spike functions.
I don't exactly know how to show this. In class we've seen that for a bounded from below, smooth, convex function, the functional is weak lower semicontinuous. But here $u$ doesn't have to be smooth or convex. I've also found an approach with an epigraph argument here on stackexchange, but we haven't seen this argument in class so I suppose I should show this with the definition of weak lower semicontinuous and some inequalities. I thought of Poincare inequality but I still wasn't able to show the assumption.
Could someone give me a hint on how to show the lower semicontinuity of this functional?