Let $C=\{u\in C^1[0,1]\mid u(0)=0,u(1)=1\}$. What is the minimum of the functional $F(u)=\int_0^1 |u'(x)-u(x)| dx$.
What I guess is that the minimum is attained when $u(x)=x$.
What follows is my try. Let $C_0=\{u\in C^1[0,1]\mid u(0)=u(1)=0\}$. For any function $u\in C$,one has $u-x\in C_0$. Then we only need to consider the minimization of $$ \int_0^1|v'(x)-v(x)-x| dx=\int_0^1|x+v(x)-v'(x)| dx $$ over $C_0$. But, I do not know how to process further.