Minimizing the $L^2$ norm of the derivative (Schilder's theorem)

Question

We define the following functional on $C^0([0,1],\mathbb{R})$: $$ I(f)=\begin{cases} \frac{1}{2}\int_0^1(f')^2=:\frac{1}{2}\|f'\|_2^2&\text{ if $f$ is absolutely continuous with square-integrable derivative}\\ \infty&\text{ otherwise} \end{cases} $$ and we would like to minimise this over the set $C=\{f:\int_0^1f^2\geq 1\}$, i.e. find the quantity $$ \inf_{f:\|f\|_2\geq 1}I(f)=\inf_{f:\|f\|_2\geq 1}\tfrac{1}{2}\|f'\|_2^2. $$ This problem actually comes from p.17 in "Large Deviations" by Varadhan, where we're trying to apply Schilder's theorem for large deviations for Brownian motion to the set $C$ above. Varadhan just says this is an eigenvalue problem without elaborating, but I'd welcome some help on understanding how to solve this!

Answer 1

This is mainly a consequence of Lagrange multipliers. First note that if $||f||_2 > 1$, you can rescale your energy with $f \mapsto \frac{1}{||f||_2}f$ and decrese $I(f)$ that way. So we can simplify the constraint to $||f||_2^2=1$.
For $I=(0,1)$ and $\langle \cdot, \cdot \rangle$ the standard $L^2(I)$ scalar product, the Euler-Lagrange associated to $I(f)$ is given by $$ \langle f',\phi' \rangle=0 \; \; \forall \phi \in C_c^{\infty}(I) $$ and for your constraint it is given by $$ \langle f, \phi \rangle =0 \; \; \forall \phi \in C_c^{\infty}(I) $$ Using the methods of Lagrange multipliers, you know that a solution of your variational problem satisfies $$ \langle f',\phi' \rangle + \lambda \langle f,\phi \rangle=0 \; \; \forall \phi \in C_c^{\infty}(I) $$ for some $\lambda \in \mathbb{R}$. Now you can apply regularity theory to deduce that $f \in C^2(I)$ and then integrate by parts to arrive at $$ \langle f''-\lambda f, \phi \rangle=0 \; \; \forall \phi \in C_c^{\infty}(I) $$ from which follows $$ f''=\lambda f $$ This is an eigenvalue problem for the linear operator defined by $Tf=f''$.
Solving this problem can be done via abstract existence theory for eigenvalues or the direct method of the calculus of variations - there could also be a more elementary approach to this problem which I can not recall as of now. There are many routes to take, just pick one that you are familiar with.