Non convex optimization problem in $W_0^{1,2}$ 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?
 A: Perhaps not the answer you are expecting, but here is my modest insight. For the sake of notation let me denote $J(\alpha)$ the value of the infimum, which is actually a minimum. (The direct method works in CoV here)

*

*As you pointed out, one has a lower bound $J(\alpha)\geq \lambda_0$, where $\lambda_0$ is the Dirichlet-Laplacian principal eigenvalue. Here you can even compute $\lambda_0$ explicitly, since in $[0,\pi]$ the principal eigenfunction is given by a suitable multiple of $\sin (x)$ and therefore $\lambda_0=1$  since $-\sin ''=\sin$.
Hence
$$
1\leq J(\alpha)
$$


*For an upper bound, let $ (u_1(x),\mu_1)$ be the Dirichlet-Laplacian  principal eigenfunction/eigenvalue on $[0,\pi/3]$. Similarly, let $ (u_2(x),\mu_2)$ be the Dirichlet-Laplacian principal eigenfunction/eigenvalue on $[2\pi/3,\pi]$, thus with $\mu_1=\mu_2=\mu=9$
($u_1$ is a suitable multiple of $\sin(3x)$ and $u_2$ a suitable translation thereof, hence $\mu=3^2=9$.)
Let now $\tilde u_1,\tilde u_2$ be the renormalizations of $u_1,u_2$ such that $\int_{0}^{\pi/3}\tilde u_1^2=\int_{2\pi/3}^\pi \tilde u_2^2=\frac 12$.
Then by standard properties of truncation in $H^1$ the function
$$
u(x):=\begin{cases}
\tilde u_1(x) & \mbox{if }x\in[0,\pi/3]\\
0 & \mbox{if }x\in[\pi/3,2\pi/3]\\
\tilde u_2(x) & \mbox{if }x\in[0,\pi/3]
\end{cases}
$$
belongs to $H^1_0$, has mass $\int_0^\pi u^2=1$, and trivially satisfies the interior mass constraint as $\int_{\pi/3}^{2\pi/3}u^2=0\leq \alpha$.
As a consequence this function is an admissible minimizer, hence
$$
J(\alpha)\leq \int_0^\pi |u'|^2
=\int_0^{\pi/3} |\tilde u_1'|^2  +  0   + \int_{2\pi/3}^\pi |\tilde u_2'|^2
=9
$$
(since we renormalized to half masses, each left and right piece $\tilde u_1,\tilde u_2$ contributes with a factor $\mu_i\int \tilde u_i^2=9\times\frac 12$).

As a conclusion the objective functional $J(\alpha)\in[1,9]$ does not blow up as $\alpha\to 0$. The natural conjecture here is that the construction in my step 2 is plausibly close to optimal, i.e.
$$
J(\alpha)\to 9\qquad\mbox{as }\alpha\to 0\qquad ???
$$
(at least this is my intuition, but maybe there's more to it than meets the eye)
So maybe I misunderstood your quetion: perhaps you already realized all of that, in which case the question should rather read: how fast is this convergence? (i.e. can we find a lower bound for $9-J(\alpha)\geq 0$?)
A: 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.
