Riesz-type inequalities and maximum flux exchange flow

TL;DR

This paper investigates shape optimization problems related to flux exchange flow in a duct, providing solutions in one dimension and exploring symmetry and existence of optimizers in a mathematical framework involving the Dirichlet Laplacian.

Contribution

It determines the optimizers for a shape optimization problem involving the Green operator in one dimension and applies these results to flux exchange flow, also establishing existence and symmetry properties.

Findings

Explicit optimizers identified in 1D case.

Existence of solutions for relaxed optimization problem.

Symmetry properties of the optimizers derived.

Abstract

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.