Suppose that $U \subset \mathbb{R}^2$ is nonempty, open, connected and bounded. Consider a Poisseuille flow in the pipe $U \times \mathbb{R}$. That is: a time-independent incompressible flow of the form:
$$v:U \times \mathbb{R} \rightarrow \mathbb{R}^3: (x,y,z) \mapsto (0,0,w(x,y))$$
which satisfies:
$$\frac{\partial^2w}{\partial x^2} + \frac{\partial^2w}{\partial y^2} = k < 0$$
$$w \left.\right|\_{\partial U} = 0$$
with $k$ some constant involving the pressure-gradient along the pipe's axis and the viscosity.  

If we consider pipes with fixed cross-section,
$$\int\_U \mathbb{d}x \mathbb{d}y = C , $$
is it true that the flow rate $\int\_U w(x,y) \mathbb{d}x \mathbb{d}y$ is maximized only if $U$ is a disc? 


If the answer is negative, consider instead pipes with a fixed boundary length,
$$\int\_{\partial U} \mathbb{d}l = L,$$
is it now true that the flow rate is maximized only if $U$ is a disc?



Do circular pipes maximize flow rate?
The answer is positive, this is a consequence of a result by Talenti ["Elliptic Equations and Rearrangements", Annali SNS 3 (1976)]. Let $k>0$, $-\Delta u= k$ in $U$, $u=0$ on $\partial U$, and consider the ball $B$ having the same volume as $U$. If $v$ is the solution of $-\Delta v=k$ in $B$, $v=0$ on $\partial B$, then $u^\* \leq v$ in $B$, where $u^\*$ is the spherically symmetric rearrangement of $u$. Since $u$ and $v$ are positive functions, and rearrangements preserve norms, one has $\|u\|\_1 = \|u^\*\|\_1 \leq \|v\|\_1$, which is the claim.


