Riesz-type inequalities and maximum flux exchange flow

Let $D$ stand for the open unit disc in $\mathbb{R}^d$ ($d\geq 1$) and $(D,\,\mathscr{B},\,m)$ for the usual Lebesgue measure space on $D$. Let $\mathscr{H}$ stand for the real Hilbert space $L^2(D,\,m)$ with standard inner product $(\cdot,\,\cdot)$. The letter $G$ signifies the Green operator for the (non-negative) Dirichlet Laplacian $-\Delta$ in $\mathscr{H}$ and $\psi$ the torsion function $G\,\chi_D$. We pose the following problem. Determine the optimisers for the shape optimisation problem $$\alpha_t:=\sup\Big\{(G\chi_A,\chi_A):\,A\subseteq D\text{is open and}(\psi,\chi_A)\leq t\,\Big\}$$ where the parameter $t$ lies in the range $0<t<(\psi,1)$. We answer this question in the one-dimensional case $d=1$. We apply this to a problem connected to maximum flux exchange flow in a vertical duct. We also show existence of optimisers for a relaxed version of the above variational problem and derive some symmetry properties of the solutions.