Riesz-type inequalities and maximum flux exchange flow

Let $D$ stand for the open unit disc in $mathbb{R}^d$ ($dgeq 1$) 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:=supBig{(Gchi_A,chi_A):,Asubseteq Dtext{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.