To show the importance of the choice of the topology for the direct method we have been assigned the following exercise which I've not been able to solve due to my lack of understanding on how strong and weak topologies on Sobolev spaces are defined:
Consider the Sobolev space $H_0^1[(0,1)]$ and the functional $$F(u):=\int_0^1\left(|u'(t)|^2-1\right)^2+u^2(t)\thinspace dt$$
Show that:
- $F$ is lower semi-continuos wrt the strong $H_0^1$ topology
- $F$ is not semi-continuos wrt the weak $H_0^1$ topology
- minimizing sequences are precompact wrt the weak $H_0^1$ topology
- minimizing sequences are not precompact wrt the strong $H_0^1$ topology
As a hint he sketched the following graph which suggests a sequence of functions in $H_0^1$ converging to the zero function (pointwise) but whose derivatives have modulo $1$ $\mathcal{L}^1$-a.e (where they are defined).
Intuitively it is clear that this sequence of functions is a minimizing sequence as $u_n\searrow 0$ and $u_n'=1$ (in the above sense) so that the functional converges to $0$, but how to formalize this? In particular what is to be understood under these notions of strenght of the topology?
I hope to have been clear enough, many thanks in advance.