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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?

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It does not. Take real $\beta > 0$ and $u(x) = x^\beta.$ Your functional becomes $ \frac{\beta^2}{2 \beta + 1},$ which goes to $0$ as $\beta \rightarrow 0^+.$ However, you already know that the functional cannot attain the value $0.$