Minimizing an integral operator

Question

I'm having some trouble in showing wether or not the following minimum exists \begin{equation} \min\{\int_0^1\arctan(\dot{u}^2(x) +u^2(x))dx\colon \quad u\in\mathscr{C}^1([0,1]),\quad u(0)=1\} \end{equation} my intuition suggests me that, since the analogous problem without $\arctan$ has a minimum, this should have it too, but I'm not able to use the direct method.

Answer 1

As already pointed out in the comments, the functional can be made arbitrarily small hence its infimum is $0$. Also observe that for any continuous positive function $f$, if there exists an $x$ s.t. $f(x)>0$ then $f(x)>0$ on a an entire nhd, and hence its integral will not be zero. Since your function space consists of functions satisfying $u(0)=1$ then the composition of $arctan$ with $u'^2 +u^2$ certainly satisfies this criterion. Hence if we denote your functional by $S$ then $S(u)>0$ for all u in your function space and the infimum is never achieved for any particular $u$ in your function space. So the minimum doesn't exist. This problem is basically analogous to finding the minimum of the function $f(x) = x$ on $(0,1)$. Notice the infimum is $0$ but no element in the domain of the function actually evaluates to $0$.