Consider the Bolza problem
$$ \inf\left\{F(u)=\int\limits_0^1 ((1-u'^2)^2+u^2)\, dx, u\in W^{1,4}(0,1), u(0)=0=u(1)\right\}. $$
Show that $\inf F(u)=0$, but that it does not exist an $u_0$ with $F(u_0)=0$.
Hello! To my opinion the fist step is to show that $F(u)\geq 0~\forall~u$. This is easy, I think, because the integrand is always $\geq 0$ anf therefore the integral, which means $F(u)$.
Then I have to find a sequence with $\lim\limits_{n\to\infty}F(u_n)=0$.
Can anybody help to find such a sequence? I did not have an idea yet...
How can I construct such a sequence? I don' t see that.
Thank you very much for helping!
The first part is OK. For the second part, if there is a $u \in W^{1,4}(0,1)$ with $u(0)=0=u(1)$ such that $F(u)=0$. Then $u^2=0=1-(u')^2$ a.e. in $(0,1)$, i.e. $u=0$ a.e. in $(0,1)$, but $u'\ne 0$ a.e. in $(0,1)$. This is not possible, hence there is no such a $u$.