Consider $$ F(v):=\int\limits_0^1\lvert v'(x)\rvert^2\, dx, $$ with $$ \left\{v\in H^{1,2}(0,1), v(0)=0=v(1)\right\}. $$
Show that a minimum exists and use the direct method.
When I am right informed, I have to show that:
1.) $H^{1,2}(0,1)$ is reflexive.
2.) $F$ is coercive.
3.) $F$ is weakly lower semi continiuos.
Is that right?
The problem as it stated is to simple, because is easilty to verify that $v=0$ is the minimum. So I am gonna fulfill te request of math12 in the comments.
Let $\Omega\subset\mathbb{R}^N$ be a bounded open set.
1 - An functional that satisfies 1,2,3.
Let $f\in L^2(\Omega)$. Define $F:H_0^1(\Omega)\rightarrow\mathbb{R}$ by $$F(u)=\frac{1}{2}\int_\Omega|\nabla u|^2-\int fu$$
The functional $F$ is called the energy functional associated with the problem $$\tag{1} \left\{ \begin{array}{rl} -\Delta u =f&\mbox{in $\Omega$} \\ u=0 &\mbox{in $\partial\Omega$ } \end{array} \right. $$
The minimum of $F$ is a wealy solution of (1).
2 - An Functional that not satisfies all yoru hypothesis.
This is a more general functional thats contains the first one
Let $\lambda_1$ be the first positive eigenvalue of the Dirichlet Laplacian, i.e. there exist $u\neq 0\in H_0^1(\Omega)$ such that (in the weak sense)$$\tag{2} \left\{ \begin{array}{rl} -\Delta u=\lambda_1u &\mbox{$\Omega$} \\ u=0 &\mbox{$\partial\Omega$} \end{array} \right. $$
and if $\lambda<\lambda_1$ satisfies (2) then $u=0$
Again let $f\in L^2(\Omega)$ and define $F:H_0^1(\Omega)\rightarrow\mathbb{R}$ by $$F(u)=\frac{1}{2}\int_\Omega |\nabla u|^2-\lambda\int_\Omega u^2-\int_\Omega fu$$
If $\lambda\geq\frac{\lambda_1}{2}$ then this problem is not coercive. On the other hand, if $\lambda<\frac{\lambda_1}{2}$ the problem satisfies 1,2,3.
Notes: If you want to understand these problems, you have to understand some things first.
I - In $H_0^1(\Omega)$ the number $(\int_\Omega |\nabla u|^2)^{\frac{1}{2}}$ is a norm in $H_0^1(\Omega)$ equivalent to the usual norm. Moreover, you have Poincare inequalitu $$\int_\Omega u^2\leq\frac{1}{\lambda_1}\int_\Omega |\nabla u|^2$$
II - I suggest you to take some books of functional analysis (like Brezis for example) to understand why $H_0^1$ is reflexive and why the norm is weakly sequentially lower semicontinuous.