Given the functional $\Phi(u) \; = \; \int_0^1 x^2.|u'(x)|^2 dx,$
I am looking for the infimum in class $C : u\in C^1(0,1) \cap C^0[0,1]$ with end point values at $u(0)=0 $ , $u(1)=1.$
The function inside integral is $\ge0$ , hence $u(x)=c$ for some constant gives a $0$ value for the functional $\Phi(u)$ but this $u$ doesn't satisfy the end values . On the other hand $u(x)=x$ satisfies the end point values.
So I am wondering if at all it possesses a minimiser. I need some hints to argue if $\Phi$ attains its infimum?