I want to show that there is no minimizer for $F[u]:=\int \limits_{0}^{1} \sqrt{u(x)^2+u^{\prime}(x)^2}dx$ in $\mathcal{Z}:= \lbrace u \in W^{1,1}(0,1):u(0)=0 \; \; \text{and} \; \; u(1)=1 \rbrace $.
The idea is to show at first that $F[u]\geq 1$ for all $u \in \mathcal{Z}$ and than find a series $u_k \in \mathcal{Z}$ with $ \lim_{k \rightarrow\infty} u_k = 1 $.
I need some help for the two steps. I don't know how to determine the following Integral.
$F[u]\geq \int \limits_{0}^{1} |u^{\prime}(x)|dx$
At the next step i tried some linear functions and also this one:
$u_k = 0$ for $x \in (0,\frac{k-1}{k}]$
$u_k=\frac{e^{k(x-1)+1}-1}{e-1}$ for $x \in (\frac{k-1}{k},1)$
Thanks in advance.