Let $0< \alpha \ll 1$. I'm trying to minimize $\int\_0^\pi |f'|^2 dx$ over the functions $f \in W\_0^{1,2}([0,\pi])$ (or at least find "good" lower bound in terms of $\alpha$) such that satisfy the following constraints:
$$
\begin{cases}
\int\_0^\pi f^2 dx = 1, \\
\int\_{\pi/3}^{2\pi/3} f^2 dx \leq \alpha.
\end{cases}
$$
In other words, I'm trying to find a function that vanishes at $0$ and $\pi$, has most of its mass near the end points, and minimizes its Dirichlet energy.


I do have a trivial lower bound given by Poincare (Wirtinger) inequality which is not in terms of $\alpha$.
I don't know where to start studying this problem.


I have tried taking Fourier transform of $f$ but the second constraint gets complicated. To be honest, I started with the formulation in the Fourier space which is the following:
$$
\begin{cases}
\mbox{find sequence $(a\_n)\_n$ that minimizes} &\sum\_n na\_n^2, \\
\mbox{subject to:} &\sum\_n a\_n^2 = 1, \\
& \int\_{\pi/3}^{2\pi/3} (\sum\_n a\_n sin(n x))^2 dx \leq \alpha.
\end{cases}
$$


Does anyone know where to start or any related reference?



Non convex optimization problem in $W_0^{1,2}$
You can treat this as a problem with two Lagrange multipliers. Then by standard methods, a minimizer $f$ has to exist (by convexity in $f'$) and has to be a weak solution to
$$-f'' + \lambda f + \mu f \chi\_{[\pi/3,2\pi/3]} = 0$$
with $\lambda, \mu \in \mathbb{R}$ and $\mu \leq 0$ because the constraint is one-sided.


Solving this equation on its sub-intervals and using the boundary condition gives you
$$ f(x) = \begin{cases} a\_1 \sin(\lambda x) &\text{ for } x \in [0,\pi/3] \\ b\_1 \sin((\lambda + \mu)x) +b\_2 \cos((\lambda + \mu)x) &\text{ for } x \in [\pi/3,2\pi/3] \\ a\_2 \sin(\lambda (x-\pi)) &\text{ for } x \in [2\pi/3,\pi]. \end{cases} $$


Using a bit of regularity theory on $f'' = - \lambda f - \mu f \chi\_{[\pi/3,2\pi/3]} \in L^2([0,\pi])$ gives you $f \in W^{2,2}([0,\pi])$, so since we are in 1d, $f'$ is absolutely continuous. This gives you two equalities at $\pi/3$ and $2\pi/3$ each.


You also can explicitly calculate $\int\_0^\pi f^2 = 1$ and have $\int\_{\pi/3}^{2\pi/3} f^2 = \alpha$ or $\mu = 0$.


This gives you a system of 6 (independent, if I am not mistaken) equations on 6 variables. I will not try to solve it here, but my guess would be that it is a tedious but doable task. You will likely end up with a countable family of solutions, but selecting the smallest should be easy.


