Riesz-type inequalities and maximum flux exchange flow

Let $D$ stand for the open unit disc in ${\mathbb{R}}^d$ ($d\ge 1$) and $(D,\mathcal{B},m)$ for the usual Lebesgue measure space on $D$. Let $\mathcal{H}$ 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 $-\mathrm{\Delta }$ in $\mathcal{H}$ and $\psi$ the torsion function $G{\chi }_D$. We pose the following problem. Determine the optimisers for the shape optimisation problem $${\alpha }_t:=\sup \{(G{\chi }_A,{\chi }_A):A\subseteq D\text{is open and}(\psi ,{\chi }_A)\le t\}$$ 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.